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ELEMENTARY    ALGEBRA 


FIRST   COURSE 


BY 


JOHN   C.   STONE,   A.M. 

HEAD    OF    THE    DEPARTMENT    OF    MATHEMATICS,    STATE    NORMAL    SCHOOt 

MONTCLAIR,    NEW    JERSEY,    CO-AUTHOR    OF    THE    80UTHWORTH-8TONB 

ARITHMETICS,    THE    8TONE-MILLIS   ARITHMETICS,    SECONDARY 

ARITHMETIC,    ALGEBRAS,    GEOMETRIES,    ETC. 

AND 

JAMES   F.    MILLIS,   A.M. 

HEAD   OF    THE    DEPARTMENT   OF   MATHEMATICS,    FRANCIS  "MT.    PARKB9. 

SCHOOL,  CHICAGO,  CO-ADTHOR  OF  THE  8TONE-MILLIS  ARITHMETICS, 

SECONDARY    ARITHMETIC,    ALGEBRAS,    AND   GEOMETRIES 


^v  TToXX'  dXX.a  TToXv 


BENJ.   H.   SANBORN   &   CO. 

CHICAGO  NEW  YORK  BOSTON 

1915 


'■"?  3ii 


CJOPTBIGHT,  1911, 
BY 

JOHN  C.  STONE  and  JAMES  F.  MILLIS 


i 


PREFACE 

The  work  in  this  book  is  that  comprised  in  the  usual  first-year 
course  in  algebra  in  secondary  schools.  The  material  is  so 
arranged  as  to  adapt  the  book  to  the  use  of  those  schools  that 
give  only  a  brief  course  in  the  subject,  as  well  as  schools  giv- 
ing more  extended  or  more  difficult  courses.  A  more  advanced 
book,  a  companion  to  this,  called  the  Second  Course,  reviews 
the  important  topics  covered  in  the  First  Course,  amplifies  the 
treatment  of  these  topics  where  necessary,  and  covers  all  addi- 
tional topics  that  are  required  for  college  entrance. 

In  the  preparation  of  this  book,  the  authors  have  combined 
what  teachers  have  found  to  be  the  permanently  valuable  features 
of  their  Essentials  of  Algebra  with  those  features  which  charac- 
terize the  most  modern  ideas  in  the  teaching  of  secondary  school 
mathematics,  and  which,  through  widespread  test  by  application 
in  the  classroom,  have  become  generally  accepted.  It  is  believed 
that  this  text  will  be  found  teachable  and  well  balanced.  It  is 
free  from  "fads"  and  untried  experiments,  yet  sufficiently  in 
harmony  with  present  tendencies  to  meet  the  demands  of  the 
most  progressive  teacher. 

The  book  contains  a  minimum  of  theory  and  a  maximum  of 
practice.  Abundance  of  drill  is  provided  throughout  the  book. 
New  processes  and  principles  are  psychologically  and  adequately 
developed,  but  they  are  finally  mastered  by  the  pupil  through 
plentiful  application  in  drill  exercises  and  problems. 

Throughout  the  text  the  work  has  been  planned  so  as  to  intro- 
duce the  pupil  to  only  one  new  difficulty  at  a  time,  and  this  is 
mastered  through  copious  drill  before  proceeding  to  the  next. 
This  controlling  principle  in  the  development  and  sequence  of 
subject  matter  is  one  of  the  fundamentally  important  features 

iii 


:lv.       './::«:;  :  preface 

.^if'ijiie^'fciook.  J  0^)serve,  for  example,  the  introductory  chapter, 
in*  which'oiie  hew  point  of  notation  is  developed  at  a  time. 
Observe,  also,  the  entire  treatment  of  the  equation. 

The  pupil  is  led  to  see  the  subject,  not  merely  as  a  system 
of  exercises  to  be  pursued  for  the  purpose  of  mental  discipline, 
although  the  authors  are  thorough  believers  in  the  mental  disci- 
pline feature  of  the  study,  but  rather  as  a  scientific  instrument 
for  solving  certain  types  of  problems  such  as  are  actually  en- 
countered in  the  world's  work.     See  §§  1,  2,  15,  16,  17,  22,  etc. 

To  this  end  adequate  use  has  been  made  of  real  applied  prob- 
lems of  the  various  types  encountered  in  practical  life.  Thus, 
in  Chapter  I  computation  by  use  of  practical  formulae  is  intro- 
duced. In  Chapter  II  the  equation  is  introduced  as  a  scientific 
instrument  used  in  the  solution  of  practical  problems.  In  Chap- 
ter III  many  practical  applications  of  negative  number  are  made, 
etc.  These  real  applied  problems  lend  intense  interest  to  the 
work.  Through  them  the  work  is  legitimately  motivated. 
Furthermore,  it  is  through  the  application  of  the  pupil's  knowl- 
edge of  algebra  in  the  solution  of  these  real  problems  that  the 
knowledge  is  led  to  function.  The  old-time  useless  puzzles  have 
been  eliminated  from  this  book. 

Algebra  has  been  correlated  with  arithmetic  and  geometry. 
Beview  and  use  of  the  various  process  with  whole  numbers, 
common  and  decimal  fractions,  have  been  made  in  the  practical 
computation  with  formulae,  and  in  the  process  of  checking  work 
by  evaluation  of  the  literal  expressions  involved.  Short  methods 
of  multiplication  of  arithmetical  numbers  are  taught  in  connec- 
tion with  algebraic  multiplication  and  factoring,  as  in  §§  75,  77, 
etc.  Principles  and  processes  in  arithmetic  have  been  made  the 
starting  points  in  the  development  of  many  principles  and 
processes  of  algebra,  as  in  §§  50,  51,  54,  etc.  Considerable  use 
has  been  made  of  percentage  in  the  solution  of  certain  types 
of  business  problems.  The  facts  of  mensuration  have  been  much 
used,  and  some  empirical  knowledge  of  simple  facts  in  geometry 
with  which  the  pupil  is  familiar  has  been  drawn  upon  in  problems. 

The  graph  has  been  used  as  a  natural  means  of  solving  certain 


PREFACE  !  ;  ;«  r;  v 

types  of  problems  and  of  interpretation  of  algebraic  principles, 
rather  than  as  an  excrescence  in  the  form  of  topics  of  anttlytio 
geometry. 

Supplementary  exercises  have  been  given  at  the  end  of  each 
chapter.  They  adapt  the  text  to  the  use  of  schools  with  different 
kinds  of  courses.  These  exercises,  as  a  rule,  are  more  difficult 
than  those  in  the  body  of  the  chapters.  Schools  wanting  a  brief 
course  may  omit  these  entirely.  They  may  be  used  for  reviews 
or  in  classes  wanting  a  more  difficult  course. 

The  authors  wish  to  acknowledge  their  indebtedness  to  all 
those  whose  timely  suggestions  and  criticisms  have  helped  in  the 
preparation  of  this  textbook,  and  especially  to  Mr.  K,.  G.  Kinkead, 
Assistant  Superintendent  of  Schools,  Columbus,  Ohio;  who 
has    read  critically  all  of  the  manuscript. 

JOHN   C.   STONE. 

JAMES  F.   MILLia 
MAY.  1011. 


•  •     •  '  •" 


CONTENTS 

CHAPTER  »»AGB 

I.    The  Formula:   General  Number 1 

II.    The  Equation 19 

III.     Positive  and  Negative  Numbers 34 

^  IV.    Addition  and  Subtraction  of  Literal  Expressions  .  60 

^^  V.    Multiplication  and  Division  of  Literal  Expressions  74 

VI.    Linear  Equations  :   Problems 92 

VII.    Special  Products  and  Quotients 117 

VIIL    Factors.   Multiples.   Equations  solved  by  Factoring  136 

<^1X.    Fractions 164 

^X.    Fractional  Equations.     Problems.    Formulae     .        .  189 

XI.    Proportion.    Variables 206 

XII.    Systems  of  Linear  Equations 232 

JXIIL    Square  Root.    Quadratic  Surds 255 

XIV.     Quadratic  Equations 269 

XV.     Systems  involving  Quadratic  Equations     .        .        ,  286 

XVI.     Exponents 294 

Miscellaneous  Exercises 300 


vB 


ELEMEI^TARY  ALGEBRA 

FIRST   COURSE 

CHAPTER   I 
THE  FORMULA:    GENERAL   NUMBER 

1.  Algebra.  —  Of  the  many  kinds  of  problems  encountered  in  the 
world's  work,  some  are  solved  by  arithmetic  alone,  some  by  algebra^ 
and  some  by  other  branches  of  mathematics.  Algebra,  like  arith- 
metic, deals  with  numbers.  There  is  no  clear  line  of  distinction 
between  these  two  subjects.  But  in  addition  to  the  Hindu  sym- 
bols of  arithmetic,  algebra  employs  letters  to  represent  numbers. 
In  addition  to  the  integers  and  fractions  of  arithmetic,  it  deals 
with  new  kinds  of  numbers.  And,  as  we  shall  see,  it  employs 
some  new  and  interesting  principles  in  the  use  of  numbers. 

2.  The  Formula.  —  Many  of  the  rules  which  the  student  has 
already  encountered  in  arithmetic,  as  well  as  practical  rules 
which  are  used  in  the  various  fields  of  manufacturing,  the  trades, 
the  sciences,  etc.,  are  expressed  often  by  formulae,  in  which  the 
numbers  are  represented  by  letters.  A  formula  expresses  a  rule 
in  a  sort  of  shorthand.  This  will  be  seen  clearly  in  the  follow- 
ing problems. 

EXERCISES 

1.  How  many  square  feet  in  a  floor  12  feet  wide  and  15  feet 
long  ? 

2.  A  concrete  walk  is  4  feet  6  inches  wide  and  24  feet  long. 
Find  the  area  of  its  surface. 

t 


2  ELEMENTARY  ALGEBRA 

3.  What  is  the  area  of  a  rectangle  3.4  inches  wide  and  7.8 
inches  long  ? 

4.  In  general,  the  rule  for  finding  the  area  of  any  rectangle 
that  is  I  feet  long  and  w  feet  wide  may 
be  expressed  by  the  formula :  Area  =  1  xw 
square  feet.  . 

Find   the   area  when    I  =  3i   feet   and 
,  w  =  2  feet.    When  Z  =  16  feet  and  w  =  2^ 

feet.    When  I  =  50  feet  and  ^y  =  4 J  feet. 
When  I  =  136  feet  and  w  =  80  feet. 

3.  Particular  and  General  Numbers.  —  In  Problem  4,  §  2,  the  let- 
ter I  is  used  to  represent  the  length  of  any  rectangle,  and  hence 
if  we  consider  all  possible  rectangles,  I  may  have  many  particular 
values.  Similarly,  if  we  consider  all  possible  rectangles,  w  may 
have  many  particular  values.  It  is  seen,  then,  that  in  express- 
ing a  rule  by  a  formula,  letters  are  used  to  represent  numbers, 
and  to  such  letters  may  be  assigned  many  particular  values. 

Compared  with  the  particular  numbers  of  arithmetic,  1,  2,  3,  4, 
etc.,  numbers  represented  by  letters  are  called  general  numbers. 
Numbers  represented  by  letters  are  also  called  literal  numbers. 

4.  Factors.  Signs  of  Multiplication.  —  The  numbers  which  are 
multiplied  to  form  a  given  number  are  called  its  factors,  as  I  and 
w  inl  X  w.  In  addition  to  use  of  the  sign  x  to  indicate  multipli- 
cation, as  in  Problem  4,  §  2,  multiplication  is  sometimes  expressed 
by  placing  a  dot  between  the  two  factors,  halfway  up  from  the 
lower  edge  of  the  number-symbols. 

Thus,  Ixw  may  be  written  also  I  •  w. 

But  unless  both  of  the  numbers  multiplied  are  represented  by 
Hindu  symbols,  the  multiplication  usually  is  expressed  by  omitting 
the  multiplication  sign. 

Thus,  the  product  I  x  w  is  written  Iw,  and  2  x  lo  is  written  2w, 

Express  in  three  ways  the  product  of  5  and  I, 


/ 


THE  FORMULA  :    GENERAL   NUMBER  3 

EXERCISES 

1.  Express  the  product  of  a  and  b;  of  2,  x,  and  y. 

Give  the  value  of  2ab  when  a  and  b  have  the  following 
values : 

2.  a  =  2,  6  =  3.  5.    a  =  10,  6  =  30.          8.    a  =  25,  6  =  4. 

3.  a  =  5,  6  =  6.  6.   a  =  40,  6  =  20.          9.    a  =  75,  6  =  6. 

4.  a  =  3,  6  =  7.  7.   a  =  50,  6  =  30.        10.    a  =  60,  6  =  3^. 

11.  Kead  Exercise  4,  page  2,  and  give  a  meaning  for  J.  =  Iw. 
Find  the  value  of  A  when  Z  and  w  have  the  following  values : 

12.  Z  =  18,  w  =  10.       14.  I  =  13^,  w  =  12.      16.  I  =  2.4,  w  =  1.5. 

13.  Z  =  12.5,  IV  =  8.      15.  I  =  6.2,  w;  =  4.         17.  Z  =  8.5,  w  =  3.2. 

18.  When  6  represents  the  base  of  a  triangle,  ^  the  altitude,  and 
A  the  area,  the  area  is  found  by  the  formula  A  =  ^bh.  Give 
the  rule  which  this  formula  expresses. 

In  the  formula  A  =  ^bh,  give  the  value  of  A  when  6  and  h  have 
the  following  values : 

19.  6  =  12,  h  =  10.      21.  6  =  20,  h  =  16.5.      23.  6  =  16,  ^  =  9.5. 

20.  6  =  14,  h  =  12.      22.  6  =  8.5,  h  =  12.        24.  6  =  12.5,  ^  =  8. 

25.  If  c  is  the  circumference  and  d  the  diameter  of  a  circle, 
then  c  =  ttcZ.  Compute  the  circumference  of  a  circle  from  the  for- 
mula c  —  ird  when  d  =  10.     (tt  =  3.1416,  approximately.) 

2-6.  When  r  =  the  radius  of  a  circle,  c  =  2  7rr.  Find  c  when 
r  =  40 ;  r  =  60 ;  r  =  7.5. 

27.  The  volume  of  a  prism  is  found  by  multiplying  the  area  of 
the  base  by  the  height  or  altitude.  If  F=  volume,  6  =  area  of 
base,  and  h  =  altitude,  then  F=  bh.    Find  V  when 

6  =  40  and  /i  =  30 ;  6  =  48  and  ^  =  12 ;  6  =  96  and  h  =  lOJ. 

28.  If  the  base  of  a  prism  is  a  rectangle  whose  length  is  Z  and 
whose  width  is  w,  then  6  =  Iw.  Hence  bh  becomes  lich.  Then  we 
have  F=  Iwh.     Express  the  meaning  of  this  formula. 


ELEMENTARY  ALGEBRA 


In   the  formula  V=  Iwh,  find  V  when  I,  w,  and  h  have  the 
values : 


31.    I  =  36,  w 


30,  h 


18. 


29.  I  =  42,  w  =  16,  A  =  14. 

30.  /  =  38,  w  =  20,  h  =  12i  32.    /  =  42,  w  =  38,  /i  =  12. 

33.  The  volume  of  a  pyramid  is  computed  by  the  formula 
Vz=  -1 5^,  where  F=  volume,  6  =  area  of  base,  and  h  =  altitude. 
State  in  words  the  rule  which  this  formula  expresses. 

Find  V  when  6  =  12  and  h  =  4i 


34.  The  volume  of  a  cylinder  is  equal  to  the  product  of  the 
altitude  by  the  area  of  the  base.     Express  this  by  a  formula. 

35.  The  volume  of  a  cone  is  one  third  of  the  volume  of  a  cylinder 
of  the  same  base  and  the  same  altitude.  If  F  =  volume,  h  =  area 
of  base,  and  h  =  altitude,  write  the  formula 
for  computing  the  volume  of  a  cone. 

If  the  base  of  a  cone  is  36  square  inches, 
and  the  altitude  1  foot  6  inches,  find  its 
volume. 

36.  The  rule  for  finding  the  area  of  a 
circle  whose  radius  is  r  is  expressed  by  the 
formula,  A  =  ttt^.     Give  the  rule  which  this  formula  expresses. 

Find  A  when  r  =  4.     When  r  =  10. 

37.  Find  the  area  of  a  circle  whose  radius  is  6  inches. 

38.  When  there  is  a  steam  pressure  of  90  pounds  per  square 
inch  in  the  cylinder  of  an  engine,  what  is  the  total  pressure  on  a 
12-inch  piston  ?  (By  a  12-inch  piston  is  meant  a  piston  whose 
diameter  is  12  inches.) 


THE  FORMULA:   GENERAL  NUMBER  6 

5.  Powers.  —  In  Problem  36,  §  4,  r^  is  a  short  way  of  writing  rr. 
Similarly,  rrr  is  written  r^,  aaaa  is  written  a*,  10  x  10  x  10  x  10  x  10 
is  written  10^,  etc.  In  general,  when  all  of  the  factors  of  a  product 
are  equal,  the  product  is  called  a  power  of  one  of  the  factors,  and 
is  written  in  this  abbreviated  form. 

x^  is  read  "  x  square,"  or  "  x  second  power  " ; 
a^  is  read  "  x  cube,''  or  *'  x  third  power  " ; 
a^  is  read  "cc  fourth  power  " ; 
jc*  is  read  "  x  fifth  power  " ;  etc. 

In  a  power  such  as  m^,  the  factor  m  is  called  the  base,  and  the 
number  7  is  called  the  exponent.  The  exponent  tells  how  many 
times  the  base  is  used  as  a  factor  to  make  the  power. 

EXERCISES 

1.  Name  the  bases  and  exponents  in  the  following:  a^,  y"^y 
C,  A"",  9^,  x\ 

2.  Express  with  exponents :  mmmmmm,  bbbbbbbb,  2x2x2 
X2x2x2x2x2x2x2. 

3.  Find  the  value  of  2x2x2;  of  2^;  of  3";  of  2*;  of  3*; 
of  2«. 

4.  In  the  formula,  A  =  ttt^,  find  A  when  r  =  6.5. 

5.  The  area  of  a  circle  is  found  also  by  use  of  the  formula, 
A  =  \  ircPf  where  d  is  the  diameter.  Express  this  in  words  as  a 
rule.     Find  A  when  d  =  4^. 

6.  The  area  of  the  surface  of  a  sphere  is  computed  by  the 
formula  A  =  4:7r7^,  where  ^  =  area,  r  =  radius.  Find  the  area 
if  the  radius  is  12  inches. 

7.  The  earth  is  approximately  a  sphere  whose  radius  is  4000 
miles.     Find  its  area. 

8.  The  rule  for  computing  the  volume  of  a  cone  of  which  the 
iase  is  a  circle  is  expressed  by  V  =  i  irr^h,  where  r  =  radius  of 
base  and  h  =  altitude.     Find  V  when  r=iQ  and  ^  =  4. 


6  ELEMENTARY  ALGEBRA 

9.  The  volume  of  a  sphere  is  computed  by  the  formula, 
K  =  4-  Trr^,  where  V  =  volume  and  r  =  radius.  Find  V  when 
r  =  3. 

10.  The  diameter  of  the  moon  is  2160  miles.     Find  its  volume. 

11.  The  bases  or  parallel  sides  of  a  trapezoid  are  6  inches  and 
10  inches,  respectively,  and  the  altitude  or  distance  between  them 

-  is  8  inches.     Find  the  area. 


il2.   The  area  of  a  trapezoid  is  equal  to  the 
product  of  one  half  of  the  altitude  by  the 
sum  of  the  bases.     This  is   expressed  by 

the  formula,  A  =  ^7i  (a  +  b)^  where  h  is  the 
altitude  and  a  and  6,  respectively,  are  the  bases.  Find  A  when 
h  =  4:  feet,  a  =  3  feet,  and  6  =  6  feet. 

6.  Use  of  Parentheses.  —  In  the  formula  in  Problem  12,  §5, 
the  symbols  (  ),  which  inclose  a  +  6,  or  the  sum  of  the  bases,  are 
called  parentheses.  These  symbols  are  used  to  inclose  two  or 
more  numbers  which  are  to  be  added  or  subtracted,  etc.,  and  indi- 
cate that  those  numbers  are  not  to  be  used  singly,  but  are  first  to 
be  combined  into  one  number  and  then  the  result  used. 

Thus,  ^h(a+b)  indicates  that  a  and  b  are  first  to  be  added,  then  the 
result  multiplied  by  ^  h.  Similarly,  x(a^  —  b^)  indicates  that  a^  —  b^,  as  one 
number,  is  to  be  multiplied  by  x ;  that  is,  we  must  first  square  the  values  of 
a  and  b  and  subtract,  then  multiply  the  difference  by  x.  And  (m4-w)(m  — w) 
indicates  that  we  are  first  to  find  the  sum  of  m  and  n,  then  their  difference, 
then  multiply  these  results. 

In  using  a  formula  containing  parentheses,  the  operations  with 
the  numbers  within  the  parentheses  must  be  performed  first, 

7.  Other  Signs  of  Grouping.  —  There  are  other  symbols  having 
the  same  meaning  as  parentheses  that  are  used  sometimes  to  in- 
close numbers.  They  are  brackets  [  ]  and  braces  {  j .  A  fourth 
symbol,  called  the  vinculum,  is  sometimes  used.  It  consists  of  a 
straight  line  drawn  above  the  numbers,  as  a-\-b,  and  is  used 
especially  in  expressing  roots  and  fractions. 


THE  FORMULA:   GENERAL  NUMBER 


Thus,  B{M-  N),  BIM-N},  B{M-  N],  and  J5  •  ilf-  iVall  indicate  the 
same  thing,  viz.  that  N  is  to  be  subtracted  from  M^  and  the  remainder  mul- 
tiplied by  B. 

All  of  these  symbols  are  called  signs  of  grouping  or  signs  of 
aggregation. 

EXERCISES 

1.  In  the  formula -4  =  ^^  (a +  6),  find  A  when  ^  =  4,  a  =  ^, 
and  6  =  7.  i  ; 

2.  Find  the  value  of  d{x  —  y)  when  c?  =  12,  a;  =  5, 2/  =  1. 

3.  Find  the  value  of  (m  —  n){in  +  n)  when  m  =  10  and  71  =  6. 

4.  Find  the  value  of  D^  JZ>  -  5j  when  Z>  =  8. 

5.  Find  the  value  of  [A -\- B^[A -  B^^A^ -{- S';\  when  A  =  5 
and  J5  =  3. 

6.  Find  the  value  of  {x-^yy  when  a;  =  4  and  y  =  2. 

7.  Find  the  value  of  5(R-{-JS)\E- Sy  when  R  =  10  and 
iS  =  2. 

8.  Find  the  value  of  (g  + 10)  -^{5-g)  when  gr  =  3. 

9.  In  the  formula,  S  =  a(t  —  ^)j  find  the  value  of  S  when 
a  =  32and«  =  3. 

10.  In  the  formula  A  =  7r(R^—  1^),  find  the  value  of  A  when 
i2  =  14  and  r  =  6. 

11.  In  the  formula  S  =  2  TrR{H-\-  R),  find  the  value  of  S  when 
i?  =  4and^=5. 

Letting  the  hypotenuse  of  a  right  triangle  =  z,  the  base  =  x,  and 
the  altitude  =  y,  it  has  been  shown  in  arithmetic  that  z  =  Var^-h^'. 
Find  the  value  of  z  when : 

12.  x=3,y  =  4:.  14.    a;  =  40,  ?/ =  30.        16.   x=12,y  =  16. 

13.  x  =  6,y  =  S,  15.    a;=15,  ^  =  20.       17.   x  =  lS,y=24:. 
In  the  formula  x  =  Vz^  —  ^,  find  the  value  of  a;  when  : 

18.  z=5yy  =  4.  20.   «  =  25,  y  =  20.        22.   z  =  50,y  =  30. 

19.  2  =  10,  2/ =  6.         21.    2  =  15,  2/ =  12.       23.   2  =  35,  2/ =  28. 


8  ELEMENTARY  ALGEBRA 

8.  Number  Expressions. — Any  number  symbol,  or  combination 
of  number  symbols,  indicating  one  or  more  of  the  operations  of 
addition,  subtraction,  multiplication,  etc.,  such  as  those  used  in 
the  formulae  in  the  preceding  sections,  is  called  a  number  expres- 
sion, or  simply  an  expression. 

Thus,  irr^^  2  J.  —  3 ^  +  C,  and  ^h(b  +  b')  are  expressions. 
An  expression  involving  one  or  more  numbers  represented  hj 
letters  is  often  called  a  literal  expression. 

9.  Terms.  —  In  the  expression  5  a^  —  2  a^  +  7  /,  the  parts  5  a^, 
2  xy,  and  7  y'^  are  called  terms.  In  general,  the  parts  of  an  ex- 
pression connected  by  the  signs  -}-  or  —  are  called  the  terms  of 
the  expression.  An  expression  such  as  6  a^,  or  10  x(m  —  n),  which 
is  not  formed  by  two  or  more  parts  connected  by  the  signs  + 
or  — ,  consists  of  one  term. 

10.  Names  of  Expressions.  —  How  many  terms  in  5aWb?  In 
2P2_Q29     InAA-B-C?     lnn^-2n''-\-3n-l? 

An  expression  which  consists  of  only  one  term  is  a  monomial. 

An  expression  which  consists  of  two  terms  is  a  binomial. 

An  expression  which  consists  of  three  terms  is  a  trinomial. 

The  name  polynomial  is  applied  to  an  expression  which  consists 
of  more  than  one  term. 

It  must  be  remembered  that  an  expression  within  a  sign  of 
grouping  is  to  be  considered  as  one  term. 

Thus,  4  a  —  (&  —3  c)  is  a  binomial.    Name  the  two  terms. 

EXERCISES 

Tell  how  many  terms  in  each  of  the  following  expressions,  and 
apply  the  name  "monomial,"  "binomial,"  etc.,  to  each: 

1.  25  A'BC.  5.  (m  +  7i)2.  ^    hq-S-- 

2.  a^-f.  6.  a*  +  b\  '  ^ 

3.  a2-f2«&  +  &l  _1_  ^    62.5  QH 

4.  irhir  +  ry  ^'  '^W  '       55^ 


THE  FORMULA:    GENERAL   NUMBER  9 

10.  m22^(y-vj.  12.    VZ+C. 

^    ,         ^v  13.    f-^y  +  Q, 

11.  .7854Wl--^)-Tr.  ^     ^^       ^^ 

^\^       lOOy  14.   o?-{h-cf. 

15.   m^+w^  +  j>^  +  2  ?/m  +  2 ?/ip+  2  np. 

16.  I^-iy  +  2DC-CP,  18.   rir^  +  rsra  +  rirj. 

17.  2xy-{x'  +  y'^  +  z\  19.    (a  +  &)(a-2'). 

20.    a,-3  +  3a;22/H-3a;y2_|_2^3^ 

11.  Evaluation  of  Polynomials.  —  The  computations  by  the 
formulae  in  the  preceding  sections  have  given  practice  in  finding 
the  values  of  expressions  of  one  term. 

In  finding  the  value  of  an  expression  containing  more  than  one 
term,  the  value  of  each  term  must  he  found  before  tlie  additions  or  sub- 
tractions  indicated  between  the  terms. 

Thus,  if  a;  =  2,  3x2  _  4a:  4-  7  =  12  -  8  +  7  =  11. 
And,  if  a  =  12,  6  =  3,  c  =  2,  and  d  =  4, 
2c(a  -  6)  -  (6  +  c)d  =  2  X  2  X  9  -  5  X  4 

=  36-20 

=  16. 
Here  the  values  of  a  —  6  and  6  +  c  are  computed  first.    See  §  6. 

EXERCISES 

1.  Find  the  value  of  2P-3Q  +  7R-2S  when  P=  10, 
Q  =  6,  i2  =  l, /S'  =  4. 

2.  Find  the  value  of  3P  +  2  MN-{-  N^  when  JW=  4,  JV=  5. 

3.  Find  the  value  of  a6  +  6c— cd— 5  when  a  =  10,  6  =  6^ 
c  =  l,  d  =  S. 

4.  Find  the  value  ofic^  +  5a^  —  2a;  —  3  when  a;  =  4. 

5.  Find  the  value  of  y^  —  oc^  when  y  =4:,  x  =  3. 

6.  Find  the  value  of  (u  —  v)^  —  (w  —  xf  when  u  =  12,  v  =  S, 
w  =  5f  x=3. 


10  ELEMENTARY  ALGEBRA 

7.  Find  the  value  of  [2p  +  3  g]^  -  llOp  -  2  g][4i)  +  g]  when 
p=l,q=4..  

8.  Find  the  value  of  VA^  -|.  ^2  ^  2  —  VC-  —  75  when  ^  =  5, 
^  =  3,  C=10. 

9.  V=i7rh(f  +  r'^  +  rry     Find  Fwhenr=  12,7-'  =  15,^=18. 

10.  In  §  10,  Exercise  8,  find  value  of  expression  when  11=  96, 
/)  =  15. 

11.  In  §  10,  Exercise  11,  find  value  of  expression  when  D  =  18, 

^==60,  i^=10,  Tr=1140. 

12.  The  volume  of  the  frustum 
of  a  pyramid  is  computed  by  the 
formula  F=i/i  (b  +  B  +  VbB), 
where  V=  volume,  7i  =  altitude, 
b  =  are^  of  one  base,  and  B  =  area  of  other  base.  Find  V  when 
h  =  6  inches,  6  =  16  square  inches,  and  5  =  49  square  inches. 

In  the  formula  V  =  ^  h(b -\- B -\-  ^bB),  find  the  value  of  Fwhen: 

13.  ;i  =  6,  &  =  16,  i?  =  25.  16.    ^  =  12,6  =  49,5  =  64. 

14.  h  =  S,b  =  9,B  =  36.  17.   71  =  16,6  =  25,5  =  36. 

15.  /i  =  10,  6  =  25,  5  =  49.  18.    ^  =  18,6  =  9,5  =  25. 

In  the  formula  V  =  ^  7rh(7^  +  ?''^  +  rr'),  find  the  value  of  Fwhen: 

19.  r  =  5,  r'  =  6,  7i  =  9.  21.    r  =  12,  r' =  15,  7i  =  18. 

20.  r  =  8,  r'  =  9,  A  =  6.  22.    r  =  10,  r'  =  16,  Ji  =  12. 

12.  Coefficients. — If  an  expression  is  separated  into  two 
factors,  either  factor  is  called  the  coefficient  (co-factor)  of  the 
other.  If  a  coefficient  is  an  arithmetical  number,  it  is  called  a 
numerical  coefficient.     Otherwise,  it  is  a  literal  coefficient. 

In  6  xy,  6  is  the  numerical  coeflBcient  ot  xy  ;  x  is  the  coefficient  of  6  ^  ; 
and  y  the  coefficient  of  6x.  In  4w(a  — 6),  what  is  the  numerical  coef- 
ficient ?    What  is  the  coefficient  of  4  (a  -  6)  ?    Oia-b? 

If  no  numerical  coefficient  is  written,  the  numerical  coefficient 
1  is  understood. 

Thus,  since  x  is  the  same  as  1  x,  the  numerical  coefficient  of  a:  isl.  Like« 
wise,  the  numerical  coefficient  of  FQ  is  1,  and  of  (m  —  n)  it  is  1. 


THE  FORMULA  :    GENERAL  NUMBER  11 

13.  Similar  Terms.  —  In  what  respect  do  2  A^  7  Ay  and  10  ^ 
differ  ?  Terms  which  do  not  differ  at  all,  or  which  differ  only 
in  their  coefficients,  are  called  like  or  similar  terms. 

Thus,  x^y,  4x2y,  and  20x^y  are  similar.  The  terms,  ax,  bx,  and  ex  are 
similar  terms  in  x. 

14.  Addition  and  Subtraction  of  Similar  Terms.  —  Just  as  the  sum 
of  2  apples,  5  apples,  and  8  apples  is  15  apples,  so  the  sum  of  2  a, 
5  a,  and  8  a  is  15  a.  And  just  as  3  men  +  4  men  =  7  men,  so  3  m  -}- 
4m  =  7m.  Similarly,  6 a;  +  9 a;  =  (6  +  9)a;,  or  15a;;  C+5C  + 
30  =  (l4-5  +  3)C,or90. 

To  add  similar  terms,  add  their  coefficients  and  to  the  result  attach 
the  common  letters  with  their  exponents. 

Just  as  12  lb. -7  lb.  =5  lb.,  so  12Z-7Z  =  5?.  And  32a;!/- 
18 02/=  (32-18) ajy,  or  Uxy,  16 a^W-^ a^h^={lh-^) a^h\  oi&a^hK 

To  subtract  similar  terms,  subtract  their  coefficients  and  to  the 
result  attach  the  common  letters  with  their  exponents. 

EXERCISES 
Find  the  sums  in  1-12: 

1.  2a,  3a,  4a.  15.   60z-f25z +  52!  +  2«=? 

2.  5a;,  a;,  2a;.  16.    12d-4d=:? 

3.  lOTF,  Sir,  8TK  17.   9ri-3ri  =  ? 

4.  P,  6P,  4P,  2P.  18.    16m'-12m'=? 

5.  ^t,t,lt,&L  19.    137rP2_57rP2  =  ? 

6.  bSyl2S,S,2S.  20.    102/-102/=? 

7.  4A;,  2A;,  9A;,  5A;.  21.   25D-17D=? 

8.  6A,12AjA,A.  22.   69m;-37w;  =  ? 

9.  3a;,  5a;,  9a;.  23.    20-8  +  10  =  ? 

10.  mn,  4  mn,  3  mn,  10  mn.  24.  12  +  3  —  9  =  ? 

11.  abc,  9  abc,  3  abc,  abc.  25.  16  —  2  —  12=? 

12.  4  J5;,  3^,  15^,  10^.  26.  6n-2  n -f-3n -4w  =i? 

13.  2aH-6a-h4a  =  ?  27.  12t -^2t -7  t-t  =  ? 

14.  46  +  &  +  126=?  28.  37rZ)  +  57rZ>- 7rZ)-4,ri?=? 


12  ELEMENTARY  ALGEBRA 

29.  Compare  the  values  of  3  +  5  and  5  +  3.     Of6a  +  3a  and 
3  a  +  6  a.    Of  4  P+7  P+2  P  and  7  P+2  P+4  P,  or  2 P+4 P+7  P. 

30.  Compare    the    values   of    5  +  2  +  6    and    5 +  (2 +  6).     Of 
6^+4^+2iy  and  6^+ (4  i/+2iJ). 

Combine  the  similar  terms  in  the  following: 

31.  3a  +  2a  +  4  +  7. 

Solution.     3a  +  2a  =  5a;  4  +  7  =  11. 

Hence,  3a  +  2a  +  4  +  7  =  5a  +  ll. 

32.  5^-2J\^+9-3.  36.  10m- 8 +  10 -8  m. 

33.  122  +  4  +  6:3  +  2.  37.  3P-12-P  +  20. 

34.  9P-3  +  2P  +  7.  38.  ^+10  +  7iir-3-4/f-6 

35.  20  +  v  +  5  +  4'V.  39.  12s  +  18-s-12-5«. 

SUPPLEMENTARY  EXERCISES 

Evaluate  the  following 

1.  ^  =  Iwj  when  I  =  16  and  w  =  14. 

2.  A  =  ^  bh,  when  6  =  18  and  h  =  141. 

3.  C=  ird,  when  d  =  16  and  tt  =  3|.* 

4.  0  =  2  irr,  when  r  =  12,  and  tt  =  3|. 

5.  A  =  in^,  when  r  =  9  and  tt  =  3.1416. 

6.  ^  =  J  Trd^,  when  d  =  8  and  tt  =  3.1416. 

7.  ^  =  4  ir?-^,  when  r  =  16,  and  tt  =  3|. 

8.  ^  =  Trd^  when  d  =  25  and  tt  =  3.1416. 

9.  F=  i  irr^h,  when  r  =  6,  /i  =  12,  and  tt  =  3.1416. 

10.  F=  I  Trr^,  when  r  =  8  and  tt  =  3.1416. 

11.  A  =  \h{a  +  b),  when  ^  =  8,  a  =  10,  and  b  =  15. 

12.  V=ih{b -{-B  +  VbB),  when  /i  =  16,  &  =  25,  and  5  =  49. 

13.  F=47r^(r2  +  r'2  +  rr'),  when  7r=3i,  /i=10,  r  =  6,  and/=9. 

14.  Eecall  the  rules  you  studied  in  arithmetic,  and  give  the 
meanings  of  as  many  of  the  above  formulae  as  possible. 

*  In  practical  work  3|,  instead  of  3.1416,  is  often  used  for  the  value  of  tt. 


THE  FORMULA  :    GENERAL  NUMBER  13 

In  the  formula  for  simple  interest,  i=zj)rt,  find  i  when: 

15.  i)  =  $500,  r  =  ^%,  t  =  2.  18.  p  =  ^900,  r  =  U%,t  =  3. 

16.  p  =  $1650,  r  =  6%,  «  =  3f        19.  p  =  $1200,  r  =  5  %,  i  =  If 

17.  p=$780,  r  =  5i%,^  =  2|.       20.  i)  =  $1500,  r  =  5%,  «  =  f. 

21.  The  sum  to  which  p  dollars  placed  at  compound  interest 
will  amount  in  t  years  at  the  rate  per  cent  r  is  ^j(1  -f  r)'. 

To  what  sum  will  $  500  amount  if  placed  at  compound  interest 
for  2  years  at  6  %  ?     For  3  years  ? 

22.  To  what  sum  will  $2000  amount  if  placed  at  compound 
interest  for  3  years  at  5  %  ? 

23.  I  invest  $  100  in  the  stock  of  a  building  and  loan  associa- 
tion which  pays  7  %  interest  compounded  annually.  To  what  will 
it  amount  in  3  years  ? 

24.  The  number  of  ways  that  a  committee  of  3  persons  may  be 

selected  from  a  group  of  n  persons  is  ^  J  .    In  ^ow 

6 

many  ways  may  a  committee  of  3  be  selected  from  4  persons  ? 
From  5  persons  ?     From  6  persons  ? 

25.  If  n  gymnastic  exercises  may  be  taken  in  any  order,  the 
total  number  of  different  ways  in  which  they  may  be  selected  to 
follow  each  other  is  w(?i  —  l)(?i  — 2)  •••3  •  2  •  1.  In  how  many 
different  ways  may  a  series  of  5  exercises  be  arranged  ?  6  exer- 
cises ?     7  exercises  ? 

Note.  — The  expression  n(n  —  1)  («  —  2)  •  •  3  •  2  •  1  means  for  any  number 
n,  the  product  of  this  number  and  each  consecutive  number  to  1.  Thus,  if 
n  =  4,  the  expression  becomes  4x3x2x1. 

26.  A  party  of  8  people  secure  a  row  of  seats  at  a  theater.  In 
how  many  different  ways  may  they  be  seated  ?    (See  Problem  25.) 

27.  The  area  of  an  equilateral  triangle,  each  of  whose  sides  is 
s,  is  found  by  the  formula  A  =  \s^  V3.  Find  A  when  s  =  8. 
(V3  =  1.732). 

28.  By  drawing  the  figure  of  a  race  track  with  two  straight 
parallel  sides  S,  and  with  semicircular  ends  each  with  radius  r, 


14  ELEMENTABY  ALGEBRA 

you  will  see  from  your  knowledge  of  arithmetic  that  the  distance, 
J),  around  the  track  is  expressed  by:  D  =  2(S -\-7r 7^).  Find  D 
when  S  =  1  mi.  and  r  =  ^  mi. 

29.   In    a    figure    (polygon)    with  n    equal 
angles,  the  number  of  degrees  in  each  angle 

is  5^-11 — I.    How  many  degrees  in  each  of 

n 

the  equal  angles  of  a  triangle  ?     Of  a  figure 
with  5  equal  angles  (pentagon)?     Of  a  figure 


with  6  equal  angles  (hexagon)  ?  Of  a  figure  with  12  equal 
angles  ? 

SUPPLEMENTARY  VOCATIONAL  FORMULJE 

Note.  —  The  following  exercises  are  added  for  those  who  wish  to  give  a 
more  extensive  course  in  the  evaluation  of  formulce.  It  is  not  intended  that 
the  teacher  should  make  any  attempt  to  explain  the  meaning  of  any  of  them. 
The  uses  of  the  various  formulae  are  stated  in  the  belief  that  students  will  be 
more  interested  in  the  process  of  evaluation  if  they  know  that  such  a  thing 
must  be  done  by  people  doing  the  world's  work.  They  may  be  omitted  with- 
out interfering  with  the  subsequent  work. 

F 

1.  The  formula  (7  =  —  is  much  used  in  work  with  electricity. 

Compute  C  Aen  E  =  20.5  and  B  =  16.75. 

2.  The  velocity  of  the  recoil  of  guns  is  computed  by  the  for- 
mula F=— ,  where  F=  velocity  of  recoil,  Tr=  weight  of  gun 

W 
and  carriage,  in  pounds,  w  =  weight  of  projectile,  and  v  =  muzzle 
velocity  of  projectile.     A  lO-inch  gun  on  a  battleship  fires  a  400- 
pound  projectile  with  a  muzzle  velocity  of  1600  feet  per  second. 
Weight  of  gun  and  carriage,  22  tons.     Find  velocity  of  recoil. 

/  3.  The  force  of  pressure  P  of  the  wind,  in  pounds  per  square 
foot,  is  computed  from  P=  .005  FV  where  F=  velocity  of  wind 
in  miles  per  hour.  Find  the  force  of  the  wind  when  blowing  at 
40  miles  an  hour.  What  would  be  the  total  pressure  of  this  wind 
against  the  side  of  a  house  20  feet  high  and  60  feet  long  ? 


THE  FORMULA:   GENERAL  NUMBER 


15 


4.  If  an  object,  such  as  a  brick  dislodged  from  the  wall,  starts 
from  rest  and  falls  towards  the  earth,  the  distance  that  it  will  fall 
in  a  given  length  of  time  is  computed  by  the  formula  s=^at\ 
where  s  =  distance  in  feet,  a  =  32,  and  t  =  number  of  seconds 
elapsed.  Find  the  distance  an  object  will  fall  in  1  second ;  2  sec- 
onds ;  3  seconds  j  4  seconds ;  10  seconds ;  60  seconds  ;  5  minutes. 

Note. — This  formula  holds  accurately  only  for  bodies  falling  in  a  perfect 
vacuum.  For  bodies  falling  through  the  air,  the  velocity  is  somewhat  di- 
minished by  the  resistance  of  the  air. 

6.   How  far  would  a  body  fall  during  the  sixth  second  ? 
Suggestion. — Find  the  distance  it  would  fall  in  5  seconds  and  in  6 
seconds. 

6.  How  far  would  a  falling  body  move  during  the  thirtieth 
second  ? 

7.  To  measure  temperature,  two  different  kinds  of  thermom- 
eters are  in  use:  the  Fahrenheit  and  the  Centigrade.  On  the 
former  the  freezing  point  is  marked  32°  and  the 
boiling  point  212°.  On  the  latter  these  are  marked 
0°  and  100°  respectively.  If  the  temperature  is 
read  on  a  Fahrenheit  thermometer,  the  correspond- 
ing temperature  on  the  Centigrade  thermometer  is 
computed  by  the  formula  C  =  |(F  —  32),  where 
C  =  temperature  in  degrees  on  Centigrade  scale 
and  F  =  temperature  in  degrees  on  Fahrenheit 
scale. 

When  it  is  70°  by  the  Fahrenheit  thermometer, 
what  is  the  temperature  on  the  Centigrade  ther- 
mometer?   When  64°?    When  48°?    When  80°? 

8.  The  strength  or  capacity  for  work  of  engines 
is  expressed  by  horse  power.  The  horse  power 
of    steam    engines    is    found    by    the    formula 


C    F 


100 


-212 


-17  78 


H.P. 


plan 


where  p  =  pressure  of  steam  in 


32 


I 


33000' 

pounds  per  square  inch,  I  =  length  of  stroke  in  feet,  a  =  area  of 
piston  in  square  inches,  and  n  =  twice  the  number  of  revolutions 


16  ELEMENTARY  ALGEBRA 

per  minute.  Compute  tlie  horse  power  of  an  engine  in  wliicli  a 
test  shows  p  =  95  pounds,  ^  =  30  inches,  a  =  706.8  square  inches, 
and  n  =  100. 

9.  Find  the  horse  power  of  a  steam  engine  in  which  p  =r  110 
pounds,  1  =  24:  inches,  w  =  120,  and  the  diameter  of  the  piston  is 
16  inches. 

10.  The  horse  power  of  automobile  engines  is  computed  by  the 
formula  H.  P.  =  KND(D  -  1){E  +  2),  where  K==  .197  for  com- 
mercial touring  cars,  JV=  number  of  cylinders,  i>=  diameter  of 
cylinders,  and  E  =  ratio  of  the  stroke  to  the  diameter.  What  is 
the  horse  power  of  a  4-cy Under  engine  of  a  touring  car  in  which 
the  diameter  is  4  inches  and  the  stroke  5  inches  ? 

11.  If  a  beam  L  feet  long  is  supported  at  both  ends  and  loaded 
uniformly    throughout   its    length   with    W    pounds    per    foot, 

the  greatest  bend  or  deflection  D  at 
the  middle,  in  inches,  is  obtained  from 

5  WL 

the  formula  D  = ,  where  E  and 

384  El' 

I  have  particular  values  depending  upon  the  material  used.    This 

formula  is  used  by  architects  in  designing  buildings.     If  a  beam 

in  which  E  =  30,000,000  and  J=  f  is  12  feet  long,  and  the  load 

200  pounds  per  foot,  find  the  deflection. 

12.  The  elevation  of  a  point  above  sea  level  is  obtained  by  use 
of  the  thermometer  from  the  formula  ^=513^+  f,  where  H=: 
height  in  feet  above  sea  level,  and  t  ==  difference  (in  degrees 
Fahrenheit)  between  212°  and  the  temperature  at  which  water 
boils  at  the  place  of  observation. 

The  temperature  of  boiling  water  at  a  certain  place  is  210°. 
Find  the  elevation  of  the  place. 

13.  The  relation  between  the  height  of  a  chimney 
and  the  pressure  of  draft  which  it  produces  is  given 

by  the  formula  P=if(^-^-^),  where  P 
B=  pressure   of   draft  as  measured  by  the  height  in 


§ 


THE  FORMULA:   GENERAL  NUMBER 


17 


inches  of  a  column  of  water  that  it  will  support  in  a  tube, 
H  =  height  of  chimney  in  feet,  T  =  temperature  outside,  and 
t  =  temperature  of  air  in  chimney. 

If  if  =  150  feet,  T=  50°,  and  t  =  600°,  find  P. 

14.  In  electrical  work  problems  of  the  following  kind  are  en- 
countered: An  electric  current  flowing  from  a  point  u4  to  a 
point  B  is   divided   at  A  into  three 

branches,    each    passing    through    an  ^         [A    ^  ^^ 

electric  bell,  and  the  branches  are 
united  again  into  one  current  at  B. 
The  total  resistance  R  of  the  circuit 
from    ^    to   5    is    computed    by   the 


r,roro 


,  where  r,,  r,,  and  rgare  the  respective 


formula  R  =  - 

resistances  of  the  three  branches.    If  ?*i  =  1.2  ohms,  r2  =  1.4  ohms, 
and  7'3  =  1.6  ohms,  find  the  total  resistance  R. 

Note. — Problem  14  shows  how  the  system  of  representing  numbers  by 
letters  may  be  extended  by  attaching  subscripts  to  the  letters  of  the  alphabet. 
Thus,  by  attaching  subscripts  to  the  letter  a  we  can  create  any  number  of  new 
symbols  for  representing  numbers,  as  rti  (read  a  sub  1),  aa,  as,  04,  etc.  The 
symbols  ai  and  aa  represent  distinct  and  unrelated  values,  just  as  a  and  b  do. 

Similarly,  by  attaching  superscripts  to  letters,  new  number  symbols  may 
be  formed,  as  x'  (read  x  prime),  x"  (read  x  second),  x'",  x'^,  x^,  etc.  Use 
is  sometimes  made  of  letters  of  the  Greek  alphabet,  a  (alpha),  /3  (beta), 
7  (gamma),  5  (delta),  etc. 


15.  The  horse  power  that  may 
be  transmitted  safely  by  a  certain 
kind  of  shafting  without  breaking 
or  twisting   is   computed   by   the 

formula   H.  P.  = ,    where  n  = 

.       •        64' 

number  of  revolutions  per  minute,  and  d  =  diameter  of  shaft- 
ing in  inches.  How  many  horse  power  can  be  transmitted  by 
such  a  shafting  of  4  inch  diameter,  making  76  revolutions  per 
minute?  By  one  of  5  inch  diameter  making  100  revolutions 
per  minute? 


18  ELEMENTARY  ALGEBRA 

16.    When  a  brick  arch  is  supported  by  a  tie  rod  to  keep  the 
walls  from  spreading,  the  "horizontal  thrust,"  or  stretching  force 
exerted  on  the  rod,  in  pounds  per  linear 
foot  of  arch,  is  obtained  by  the  formula     .  '  i '  i  '  '  !j^-fnHV.ij!j!v! 


F=  — ,  where  W=  weight  on  arch 

H 

in  pounds  per  square  foot,  S  =  span  of 
arch  in  feet,  H  =  rise  of  arch  in  inches.  Find  the  strain  on  the 
tie  rod  in  an  arch  on  which  the  weight  is  360  pounds  per  square 
foot,  the  span  4  feet,  and  rise  of  arch  18  inches. 

17.  The  discharge  of  a  pump  in  gallons  per  minute  is  obtained 
from  the  formula  G  =  .03264  Td^,  where  G  =  number  of  gallons, 
T=  travel  (total  distance  traveled)  of  piston  in  feet  per  minute, 
d  =  diameter  of  cylinder  in  inches.  Suppose  that  the  diameter  of 
the  cylinder  of  a  pump  is  18  inches,  that  the  stroke  of  the  piston 
is  24  inches,  and  that  it  makes  40  revolutions  per  minute.  Find 
the  discharge. 

18.  The  horse  power  of  the  pump  required  in  Problem  17  is 
H.P.  =  .00001238  Td%  where  h  =  vertical  distance  in  feet  between 
levels  of  water  at  source  and  point  of  discharge.  Find  the  horse 
power  of  the  pump  in  Problem  17  required  to  raise  the  water  to  a 
standpipe  through  a  height  of  216  feet. 

19.  By  the  specific  gravity  of  a  solid  substance,  such  as  iron,  is 
meant  the  ratio  of  the  weight  of  that  substance  to  the  weight  of 

an  equal  volume  of  water.  The  specific 
gravity  of  an  object  may  be  found  by  first 
weighing  it  in  air,  then  weighing  it  again 
when  suspended  under  water.  An  object 
seems  to  lose  weight  when  weighed  in 
water  due  to  the  buoyancy  of  the» water. 

If  the  specific  gravity  is  s,  then  s  =  — — ,  where  W=  weight 

in  air  and  w  =  weight  in  water. 

A  piece  of  glass  weighing  40  grams  in  air  weighs  24  grams  in 
urater.    Find  its  specific  gravity. 


CHAPTER  II 
THE  EQUATION 

15.  The  Equation.  —  In  solving  some  kinds  of  problems  use  is 
made  of  equations.  An  equation  is  the  statement  that  two  expres- 
sions are  equal  or  that  they  have  the  same  value. 

Thus,  2  X  6=  10,  a  +  6  =  6  +  a,  and  4  »i  —  3  =  2  n  +  5  are  equations. 

The  two  expressions  which  are  connected  by  the  sign  =  are 
called  the  members  of  the  equation. 

For  example,  in  3  P—  2  =  4  the  expression  3  P—  2  is  called  the  first  or 
left  member,  and  the  4  is  called  the  second  or  right  member. 

In  some  equations  the  members  are  equal  for  sfll  particular 
values  of  the  general  number  involved.  These  are  called  identical 
equations  or  identities. 

Thus,  in  a^  +  2  a5  +  6^  =  (a  -f  5)2  the  members  are  equal  for  all  vahies 
whatever  that  may  he  given  to  a  and  b.  When  a  =  1  and  6=2,  the  equa- 
tion becomes  9  =  9;  when  a  =  2  and  6  =  2,  it  becomes  16  =  16 ;  when 
a  =  5  and  6  =  1 ,  it  becomes  36  =  36 ;  etc. 

In  other  equations  the  members  are  not  equal  for  all  particular 
values  of  the  general  numbers  involved.  These  are  called  con- 
ditional equations  or  simply  equations. 

Thus,  5y  —  2  =  Sy  is  true  only  under  the  condition  that  y  =  1.  And 
3  ^  +  4  =  10  is  true  only  when  B  is  2. 

16.  Problems  Expressed  by  Equations.  —  In  every  problem 
solved  by  use  of  equations,  the  values  of  one  or  more  numbers 
are  unknown. 

For  example,  in  the  problem,  "  If  eggs  cost  34  cents  a  dozen,  find  the  cost 
of  5  dozen,"  the  cost  of  6  dozen  is  an  unknown  number. 

19 


20  ELEMENTARY  ALGEBRA 

To  solve  a  problem  is  to  find  the  values  of  the  unknown  numbers. 
In  solving  a  problem  algebraically  it  is  first  expressed  in  a  sort 
.  of  shorthand  by  an  equation.  The  equation  expresses  the  rela- 
tion between  the  known  and  unknown  numbers.  An  unknown 
number  is  expressed  by  a  letter,  and  later  the  value  represented 
by  the  letter  in  the  equation  is  found. 

\         Example  1.     A  man's  salary  was  increased  by  10%,  or  -^^  of  itself.     After 
.     *    the  increase  he  received,  annually,  $1980.     What  was  his  salary  before  the 
fi-^    increase  ? 
fJ     'jr       Represent  by  d  his  salary  before  the  increase. 
\^         Then  the  problem  may  be  expressed  hy  d-\-  ^-^0,  =  ^  1980. 

jf  Example  2.     Two  men  form  a  partnership  in  business  in  which  the  first 

I  I)       invests  twice  as  much  as  the  second.    Their  profits  are  $3600.    What  part 
of  the  profits  should  each  receive  in  settlement  ? 

Let  a  represent  the  amount  the  second  should  receive. 
Then  2  a  represents  the  amount  the  first  should  receive. 
Hence,  the  problem  may  be  expressed  by  the  equation 
2a  +  a  = 


To  express  a  problem  by  an  equation  we  take  the  following 
steps : 

(1)  First  read  the  problem  to  discover  what  numbers  are  uriknown. 

(2)  Let  some  letter  represent  one  of  the  unknown  numbers. 

(3)  Tlien,  from  statements  in  the  problem,  express  all  the  other 
unknown  numbers  in  terms  of  this  letter. 

(4)  Finally,  from  another  statement  of  the  problem,  form  the 
equation  between  these  and  the  known  numbers  of  the  problem. 

ORAL  EXERCISES 

1.  If  n  denotes  a  certain  number,  what  will  denote  a  number 
10  larger?  What  will  denote  one  10  less?  One  10  times  as 
large  ?     One  -Jg-  as  large? 

2.  The  rainfall  last  year  at  a  certain  place  was  6  inches  more 
than  on  the  year  before.  If  x  represents  last  year's  rainfall,  what 
will  denote  the  rainfall  of  the  year  before?  If  x  represents  the 
rainfall  of  the  year  before,  what  will  denote  that  of  last  year? 


THE  EQUATION  21 

3.  The  melting  temperature  of  glass  is  136  degrees  lower  than 
3  times  that  of  zinc.  If  t  represents  the  melting  temperature  of 
zinc,  what  will  represent  that  of  glass  ? 

4.  One  boy  sold  20  more  than  ^  as  many  papers  as  another. 
By  use  of  some  letter  represent  the  number  sold  by  each. 

5.  If  A  has  $2/,  B  I  as  much,  and  C  ^  as  much,  what  will 
represent  the  amount  that  all  three  have  ? 

6.  Of  two  candidates  at  an  election  one  was  defeated  by  362 
votes.  By  some  letter  express  the  number  of  votes  received  by 
each. 

7.  A  rectangle  is  2  inches  more  than  3  times  as  long  as  it  is 
Tade.     By  use  of  a  letter  represent  both  the  length  and  the  width. 

8.  A  merchant  sold  coffee  at  a  profit  of  20%.  If  it  cost  C 
cents  a  pound,  what  will  express  the  selling  price  ? 

9.  If  I  lend  $/)  at  5%  for  4  years,  what  will  express  the 
amount  at  the  end  of  that  time? 

10.  The  distance  from  Chicago  to  New  York  by  rail  is  about 
900  miles.  If  a  train  ran  an  average  of  40  miles  an  hour,  how 
long  would  it  require  to  make  the  run  between  the  two  cities  ? 
If  it  ran  V  miles  an  hour,  what  will  express  the  time  required  for 
the  run? 

In  each  of  the  above  problems  one  number  has  been  expressed  in 
terms  of  one  or  more  other  numbers, 

WRITTEN  EXERCISES 

Express  the  following  problems  by  means  of  equations : 

1.  One  of  two  partners  in  a  business  invests  twice  as  much  as 
the  other.     How  should  a  profit  of  %  1200  be  divided? 

i    2.   A  real  estate  dealer  sold  a  lot  for  $1500,  and  thereby  made 
a  profit  of  25  %.     What  did  it  cost  him  ? 

3.  It  requires  2240  feet  of  wire  fencing  to  inclose  a  rectangular 
piece  of  ground  that  is  three  times  as  long  as  it  is  wide.  Find  its 
width  and  length. 


22 


ELEMENTARY  ALGEBRA 


p     4.    If  I  think  of  a  number,  double  it,  and  add  8  to  the  result,  I 
get  40.     What  is  the  number  ? 

7    5.   In  the  latitude  of  Chicago,  on  the  longest  day  of  the  year, 
the  day  is  6  hr.  8  min.  longer  than  the  night.     What  is  the  length 
B  of  each? 

J\6.  In  the  triangle  ABC,  the  sum  of  the  three 
angles  is  180°.  Angle  B  is  twice  as  large  as 
angle  A,  and  angle  C  is  three  times  as  large  as 
angle  A.     How  many  degrees  in  each  angle  ? 

17.  Solving  an  Equation.  —  It  has  been  seen  that  a  problem  may 
be  expressed  by  an  equation,  in  which  the  unknown  numbers  are 
represented  by  use  of  a  letter.  It  is  evident  that  the  problem 
may  be  solved  if  the  value  of  the  letter  in  the  equation  may  be 
found.     This  is  called  solving  the  equation. 

To  solve  an  equation  is  to  find  the  particular  value  or  values  of 
the  unknown  number  that  make  the  two  members  equal.  A  par- 
ticular value  of  the  unknown  number  thus  found  is  called  a  root 
of  the  equation. 

Thus,  in  4^4-6  =  2y-\- 10,  if  y  is  2,  each  member  equals  14.  Hence,  2  is 
a  root.     The  equation  is  said  to  "be  satisfied "  when y  =  2. 

Before  proceeding  to  the  solution  of  problems,  we  must  discover 
how  to  solve  an  equation. 

18.  Axioms.  —  If  the  weights  on  the  two  pans  of  a  balance  are 
equal,  they  balance.  If  equal  weights  are  added  to,  or  taken 
from,   the    two  pans,   will   the   resulting 

weights  balance  ?  If  the  weights  in  the 
two  pans  are  made  twice  as  great,  three 
times  as  great,  etc.,  or  one  half  as  great, 
one  third  as  great,  etc.,  will  the  resulting 
weights  balance  ? 

These  facts  about  the  balance  illustrate  certain  general  princi- 
ples in  dealing  with  numbers  which  may  be  assumed.  They  are 
called  axioms,  and  may  be  stated  as  follows : 


^ 


THE   EQUATION  23 

1.  If  equal  numbers  are  added  to  equal  number Sy  the  sums  are 
equal. 

2.  If  equal  numbers  are  subtracted  from  equal  numbers,  the  re- 
mainders are  equal. 

3.  If  equal  numbers  are  multiplied  by  equal  numbers,  the  products 
are  equal. 

4.  If  equal  numbers  are  divided  by  equal  numbers  (not  zerd)^  the 
quotients  are  equal. 

An  equation  may  be  compared  to  a  balance.  The  members  of 
the  equation  correspond  to  the  weights  on  the  two  pans  of  the 
balance.  In  the  figure,  the  fact  that  the 
weights  balance  is  expressed  by  the  equa- 
tion P4-4  =  9.  Just  as  the  weights  on 
the  two  pans  may  be  equally  increased, 
decreased,  multiplied,  or  divided,  and  con- 
tinue to  balance,  so  the  members  of  an  equation  may  be  equally 
increased,  decreased,  multiplied,  or  divided,  according  to  the  above 
axioms,  and  remain  equal. 

The  use  of  these  axioms  in  solving  equations  is  shown  in  the 
following  sections. 

19,   Equations  Solved  by  a  Single  Addition,  Subtraction,  Multipli- 
cation, or  Division. 

Example  1.  — Solve  Tr+6  =  10. 

Since  TF+6  is  0  greater  than  ir,  what  must  be  done  to  TT -1-6  to  get  W? 
Subtracting  6  from  each  member,  W=  4.  Axiom  2. 

Example  2. — Solve  8  a=  7a  +  5. 

What  must  be  done  to  7  a  +  5  to  get  5  ? 

Subtracting  7  a  fix>m  each  member,  a  =  5.  Axiom  2. 

Example  3.  —  Solve  S-i  =  12. 

Since  S  —  iisi  less  than  S,  what  must  be  done  to  S-~  4  to  get  S  ? 
Adding  4  to  each  member,  S  =  10.  Axiom  1. 

Example  4.  — Solve  3  x  =  21. 

What  must  be  done  to  3  x  to  get  x  ? 

Dividing  each  member  by  3,  sc  =  8.  Axiom  4 


24 


ELEMENTARY  ALGEBRA 


Example  5.  —  Sdlve  -  =  4. 
6 


By  what  must  -  be  multiplied  to  get  n  ? 
6 

Multiplying  each  member  by  6,  w  =  24. 


^A 


Solve : 

1.  a  +  3  =  7. 

2.  P+6=20. 

3.  'V  +  l  =  8. 

4.  72  +  9=16. 
^5.  3a;  =  2a;  +  4. 

6.  6^  =  5^  +  12. 

7.  8A:  =  23  +  7fc. 

8.  9^  =  3  +  8^. 

9.  162)  =  54-152>. 

10.  5-4  =  20. 

11.  a; -10  =  15. 

12.  Jf-50  =  12. 


ORAL  EXERCISES 

13.  jj— 9  =  1. 

14.  R-2  =  Q>. 
I         15.   w-14  =  16. 

16.    x-^  =  \. 


17.  m-li  =  |. 

18.  5  2/ =  15. 

19.  4ir=24. 

20.  2a  =  18. 

21.  6Q  =  48. 

22.  7/S  =  28. 

23.  24Z>  =  72. 

24.  100^  =  125. 


Axiom  3. 


25.  12ic  =  25. 

26.  ^  =  25. 
4 

27.  l^Z. 
2S.    ^^  =  4. 


29.  3ij  =  r. 


30. 


Q_ 


31.    ^=18, 
20 


WRITTEN  EXERCISES 


Solve : 

1. 

a; +  3.25  =  5. 

9. 

^-7f  =  8i. 

V^       2. 

i)- 0.426  =  4.32. 

10. 

y-T6=T2' 

3. 

4.6^  =  3.6^  +  2.5. 

11. 

ljr'  =  6. 

4. 

P- 4.08  =  8. 

12. 

2fv  =  5f. 

5. 
a. 

2.16  m  =  .24. 
3.1416^  =  9.5. 

13. 

3i        ^ 

7. 
8. 

4^(7=7  +  310. 
^+3f  =  7TV 

14. 

^  =  41 
3f     ^- 

THE  EQUATION  25 

20.  Any  Simple  Equation  Solved.  —  Any  simple  equation  may  be 
solved  by  using  one  or  more  of  the  processes  employed  in  §  19,  as 
shown  by  the  following  examples. 

Example  1.  —  Solve  4  w  +  15  +  13  w  =  5  n  +  99. 
Adding  similar  terms,  4  n  and  13  7i,  17  »  +  15  =  5  »  +  99. 

Subtracting  15  and  5  n  from  each  member,  12  n  =  84. 

Dividing  each  member  by  12,  w  =  7. 

Notice  that  by  subtracting  15  from  each  member  we  get  the  term  free  of 
n  out  of  the  first  member,  and  by  subtracting  5  n  from  each  member  we  get 
the  term  containing  n  out  of  the  second  member. 

Check.  —  When  n  =  7,  each  member  becomes  134,  which  shows  that  the 
answer  is  correct. 

A  number  symbol  put  in  place  of  another  in  an  expression,  as 
in  checking  the  answer  in  the  preceding  example,  is  said  to  be 
substituted  for  it.  The  student  is  familiar  with  substitution  in 
the  formulae  in  Chapter  I. 

The  root  of  evei'y  equation  should  he  checked^  or  tested  for  accv^ 
racy,  by  substituting  the  value  found  in  place  of  the  unknown  number 
in  each  member  of  the  equation. 

Example  2.  —Solve  7  F-  8  +  6  V=  28  +  4  F. 

By  combining  similar  terms,  what  does  the  first  member  become  ? 

What  must  be  added  to  each  member  to  free  the  first  member  of  the  term 
not  containing  V? 

What  must  be  subtracted  from  each  member  to  free  the  second  member  of 
the  term  containing  V? 

Show  that  the  resulting  equation  is  9  F  =  36. 

By  what  must  each  member  now  be  divided  ? 

Check  the  answer  by  substituting  its  value  in  each  member  of  the  given 
equation  and  seeing  if  they  are  equal. 

Example  3.  —  Solve  2^+^  +  9=:^+ 16. 

What  is  the  least  number  divisible  by  each  denominator  ? 
By  what,  then,  must  each  member  be  multiplied  to  free  each  term  of 
fractions. 

Show  that  the  resulting  equation  is  8  a;  +  9  a;  +  108  =  10  x  +  192. 

Now  show  that  x  =  12. 

Check  by  substituting  12  for  x  in  the  given  equation. 


26  ELEMENTARY   ALGEBRA 

It  is  evident  from  these  examples  that  the  steps  in  solving  such 
equations  are  as  follows : 

(1)  If  the  equation  contains  one  or  more  fractions,  multiply  both 
members  by  the  L.  C.  M.  of  all  the  denominators  to  free  the  equation 
effractions. 

(2)  Unite  similar  terms  by  adding,  or  subtracting,  as  the  case  may 
require. 

(3)  Free  the  first  member  of  all  terms  that  do  not  contain  the  un- 
known number,  and  the  second  member  of  all  terms  that  do  contain 
the  unknown  number,  by  adding  the  same  number  to  both  members 
or  subtrojcting  the  same  number  from  both  members.  To  remove  a 
term  with  a  plus  sign  before  it,  subtract  it,  and  to  remove  a  term  vnth 
a  minus  sign  before  it,  add  it  to  both  members. 

(4)  Divide  both  members  of  the  resulting  equation  by  the  coejficient 
of  the  unknown  number. 

(5)  Check  the  work  by  substituting  the  value  of  the  root  found  in 
each  member  of  the  given  equation. 

EXERCISES 

Solve: 

1.  7a-t-3=a  +  21.  9.  6Jf-5  =  4Jf+l. 

3.  4  6  +  4  =  25  + &.  10.  52/  +  5  =  8-32/. 

3.  5P-4  =  12-3P.  11.  20P-25  =  5P  +  5. 

4.  2^-5  =  7-^.  12.  13/i  +  15  =  ll^  +  35. 

5.  3Z>-7  =  14-4D.  13.  4:^-3  =  ^. 

6.  5a;  +  3  =  3a;+9.  14.  12c- 13  + c  =  35 +  7c. 

7.  200^.-50=50^+250.  15.  13  F-7  =  5  F  +  2- F. 

8.  5  A;  — 5  =  A; +  3.  16.  3  a;  =  10  + «. 

17.  What  is  the  least  number  divisible  by  both  2  and  3  ?  By 
what  one  number  may  both  of  the  fractions  -  and  -  be  multiplied 
to  change  them  to  whole  numbers  ? 


THE  EQUATION  21 

18.  By  what  one  number  may  each  of  the  fractions  ---,  -,  and 
f»  3     4 

— ^  be  multiplied  to  change  all  of  them  to  whole  numbers  ? 

19.  What  is  6  times  4  apples?    6  times  4a?    3  times  6«? 
8  times  In?   10  times  3^? 

20.  What  is  6  times??  8  times??  10  times  ?^?  12  times —? 


Solve: 

4        3  2  6 

23.  l£  +  i?  =  ^  +  7.  28.    3Tr-^=^+13. 

3        4         6  3        2 

24.  5  +  6  =  17-?:?.  29.    10-^  =  -  +  ^  +  3. 


2  '  6  8     4  '  2 

^-17  =  0.  30.    i2-10+-  =  ^-^. 

6  7      2      14 


21.  Problems  Solved  by  Equations.  —  The  following  example 
shows  the  complete  process  of  solving  a  problem  by  use  of  an 
eqviation. 

Example.  —  At  an  election  two  candidates,  A  and  B,  together  received 
2245  votes.  A  was  elected  by  a  majority  of  286  votes.  How  many  votee 
were  cast  for  each  ? 

Let  n  =  number  cast  for  A . 

Then  n  —  285  =  number  cast  for  B. 

Hence,  n  +  n  -  285  =  2245. 

Solving,  2  n  -  285  =  2245. 

2  n  =  2530. 
n  =  1265,  number  cast  for  A. 

Hence,  «—  285  =  980,  number  CEist  for  B* 


28  ELEMENTARY  ALGEBRA 

EXERCISES 

1.  The  initiation  fee  of  a  certain  organization  is  $40,  and  the 
annual  dues  $5.  In  how  many  years  will  a  member  have  paid 
into  the  treasury  $  100  ? 

2.  It  costs  $  4  to  have  made  the  plates  for  printing  a  circular, 
and  the  cost  of  paper  and  printing  is  \  cent  a  copy.  How  many 
copies  can  I  have  made  for  $8? 

3.  One  partner  has  4  times  as  much  money  invested  in  an  en- 
terprise as  the  other.     How  should  a  profit  of  $2500  be  divided? 

4.  Two  boys  agree  to  mow  a  man's  lawn  during  a  season  for 
$  6.  One  mows  it  3  times  and  the  other  5  times.  How  should  the 
money  be  divided  between  them  ? 

j^'^-      5.    A  grocer  bought  some  eggs  at  16  cents  a  dozen.     Thirty 
L'  K  V     were  broken,  and  he  sold  the  remainder  at  18  cents  a  dozen.     He 
found  that  he  got  back  just  the  cost  of   the  whole  lot.     How 
many  did  he  buy  ? 

6.  Divide  $  75  between  A  and  B  so  that  A  shall  receive  $  3 
more  than  twice  as  much  as  B. 

7.  Two  boys  sold  45  papers.  One  sold  ^  as  many  as  the 
other.     How  many  did  each  boy  sell  ? 

't      8.   A  boy  has   $1.05  in  dimes  and  nickels.     He  has  just   as 
many  dimes  as  nickels.     How  many  has  he  of  each  ? 

9.  Two  trains  start  at  the  same  time  from  stations  450  miles 
apart,  one  running  at  the  rate  of  35  miles  per  hour,  and  the  other 
at  the  rate  of  40  miles  per  hour.  In  how  many  hours  do  they 
meet  ? 

10.  The  resistance  of  an  electrical  battery  and  the  wire  at- 
tached to  it  is  found  to  be  3  ohms.  By  connecting  to  the  battery 
a  wire  of  the  same  size  and  material  and  three  times  as  long,  the 
resistance  is  found  to  be  6  ohms.  Find  the  resistance  of  the 
battery. 

11.  The  resistance  of  a  battery  and  the  wire  attached  to  it  is 
5^  ohms.     When  a  wire  of  the  same  kind  and  4  times  as  long  is 


\ 


THE  EQUATION  f9 

:3onnected  to  the  battery,  the  resistance  is  found  to  be  16  ohms. 
Find  the  resistance  of  the  battery. 
^  A 12.    The  total  number  of  admission  tickets  to  a  circus  was  836. 
The  number  of  tickets  sold  for  adults  was  136  less  than  twice  the 
number  of  children's  tickets.     How  many  were  sold  of  each  ? 
d*"  13.   The  distance  from   Chicago  to    San  Francisco  by  rail  is 
2563  miles,  which  is  397  miles  more  than  twice  as  far  as  from 
Chicago  to  Denver.     How  far  is  it  from  Chicago  to  Denver  ? 
^^14.    In  a  class  containing  24  boys  and  girls  there  are  6  more 
girls  than  boys.     How  many  of  each  are  there  ? 
'\15.   It  takes  360  feet  of  wire  fencing  to  inclose  a  rectangular 
lot  60  feet  wide.     How  long  is  the  lot  ? 
1     16.    A  boat  steams  against  the  current  of  a  stream  at  a  speed  of 
3  miles  per  hour,  and  with  the  current  at  a  speed  of  15  miles  per 
hour.     Find  the  speed  of  the  current.  y 

17.  A  train  left  a  station  and  traveled  at  the  rate  of  30  miles/ 
per  hour,  and  three  hours  later  a  second  train  left  the  station,  fol- 
lowing on  the  same  track  at  the  rate  of  40  miles  per  hour.     In 
how  many  hours  from  the  time  that  the  first  train  started  was  it 
overtaken  by  the  second  train  ? 

18.  A  bar  40  inches  long  is  to  be  cut  into  two  pieces.  The 
shorter  piece  is  to  be  |  as  long  as  the  longer  piece.  Find  the 
lengths  of  the  pieces. 

19.  A  board  62  inches  long  is  to  be  sawed  into  two  pieces  so 
that  one  piece  is  8  inches  longer  than  the  other.  How  long  will 
each  piece  be  ? 

20.  Surveyors,  in  order  to  mark  off  a  right  angle  on  the 
ground,  sometimes  set  three  stakes  so 
that  when  a  tapeline  is  stretched  around 
them  so  as  to  form  a  triangle,  it  is  divided 
at  the  stakes  into  parts  whose  lengths  are 
as  3,  4,  and  5.  The  ancient  Egyptians 
used  this  method  in  laying  out  their 
pyramids,  etc.     If  a  100-foot  tapeline  is 


80  ELEMENTARY  ALGEBRA 

used,  find  the  lengths  of  the  parts  into  which  it  must  be  divided 
at  the  stakes. 

Suggestion.  —  Let  3  n  represent  the  number  of  feet  in  the  shortest  part. 

21.  If,  in  Problem  20,  the  surveyors  use  a  50-foot  tapeline, 
find  the  lengths  of  the  parts  into  which  it  must  be  divided  at 
the  stakes. 

22.  A  surveyor's  chain  which  contains  60  links  is  divided  into 
three  parts  whose  lengths  are  as  3,  4,  and  5,  and  used  instead  of 
the  tapeline  in  Problem  20.     How  many  links  in  each  part  ? 

Note.  —  Students  who  have  time  will  find  it  interesting  to  use  the  method 
of  Problems  20-22  in  making  out-of-door  measurements. 

23..  A  merchant  sells  tea  for  70  cents  a  pound,  and  thereby 
makes  a  profit  of  40%  on  the  cost.     What  did  it  cost  him  ? 

24.   How  much  money  must  I  invest  in  a  business  yielding 
20%  profit  in  order  that  the  investment  and  profit  together  may 
amount  to  $1000? 

25.    The  sum  of  the  angles  a,  b,  and  c  is 
180°.     If  angle  b  equals  angle  a,  and  angle  c 
equals  one-half  of  angle  a,  find  the  value  of 
—     each. 


26.  The  sum  of  the  angles  of  a  triangle  is  180°.  If  two  of  the 
angles  are  equal,  and  the  other  one  twice  as  large  as  either  of 
them,  how  many  degrees  in  each  ? 

27.  An  angle  of  75°  is  made  up  of  the  sum  of  an  angle  of  15° 
and  5  equal  angles.  How  many  degrees  in  each  of  the  equal 
angles  ? 

28.  Two  angles  are  called  complementary  when  their  sum  is 
90°.  If  one  of  two  complementary  angles  is  15°  more  than  twice 
the  other,  how  many  degrees  in  each  ? 

29.  Three  angles  just  cover  all  of  the  plane  around  a  point 
The  difference  between  the  second  and  the  first  is  60°  and  be- 
tween the  third  and  the  second  60°.     How  many  degrees  in  each  ? 


THE  EQUATION  31 

30.  If  a  polygon  has  n  sides,  the  sum 
of  all  of  its  angles  is  180°  n- 360°. 
How  many  sides  has  a  polygon  the  sum 
of  whose  angles  is  720°  ? 

31.  Whole  numbers  such  as  5  and  6 
are  consecutive  whole  numbers.      The  sum  of  two  consecutive 
whole  numbers  is  71.     What  are  the  numbers  ? 

32.  The  sum  of  two  consecutive  odd  numbers  is  32.  What 
are  the  numbers  ? 

33.  One  number  is  twice  as  large  as  another.  If  I  take  4 
from  the  smaller  and  16  from  the  larger,  the  remainders  are 
equal.     What  are  the  numbers  ? 

34.  A  number,  its  half,  its  third,  and  its  fourth  make  100. 
Find  the  number. 

35.  In  the  papyrus  written  by  Ahmes,  the  Egyptian,  about 
1700  B.C.,  the  unknown  number  was  called  "  hau."  This  ancient 
book  contained  the  following  problem :  "  Hau,  its  \,  its  whole, 
it  makes  19 ; "  i.e.  \  of  the  unknown  number  and  the  unknown 
number  make  19.     What  was  the  unknown  number  ? 


SUPPLEMENTARY  EXERCISES 

1.  A  certain  number  exceeds  n  times  the  number  x  by  m. 
What  will  express  the  number? 

2.  D  dollars  are  invested  at  p  per  cent  interest.  What  will 
represent  the  sum  to  which  this  will  amount  in  1  year?  In  3 
years  ?     In  ^  years  ? 

3.  A  merchant  sells  sugar  costing  c  cents  a  pound  at  a  profit 
of  12%.  What  will  represent  the  selling  price?  If  it  sells  at 
a  profit  of  p  %,  what  will  represent  the  selling  price  ? 

4.  Solve  ^  +  |  =  2i. 


32  ELEMENTARY  ALGEBRA 

6.   Solve— +  — =  — +  6J. 

6.   Solve  7  =  !-?!  +  !• 
6       9       4 

7.  I  paid  $6  for  an  advertisement  of  6  lines,  as  follows: 
20  cents  a  line  for  the  first  insertion,  10  cents  a  line  for  each  of  the 
next  6  insertions,  and  2  cents  a  line  after  that.  Find  the  number 
of  insertions  at  2  cents  a  line. 

8.  How  much  per  bushel  must  a  merchant  pay  farmers  for 
wheat  in  order  to  market  it  at  $  1.14  and  make  a  profit  of  4  %  ? 

9.  How  much  per  bushel  must  a  wholesale  house  pay  for 
potatoes  in  order  to  sell  them  to  retailers  at  53  cents  and  make  a 
profit  of  6  %  ? 

10.  How  much  must  a  country  merchant  pay  farmers  for  tur- 
keys in  order  to  make  6  %,  if  he  can  get  17  cents  per  pound  foi 
them  in  the  city  market  ? 

11.  In  the  city  market,  on  a  certain  date,  unwashed  wool  sold 
for  25  cents  per  pound.  How  mucTi  per  pound  must  a  wool  buyer 
pay  farmers  for  their  unwashed  wool  in  order  to  make  a  profit  oi 
10%  when  selling  it  at  this  price,  if  it  costs  him  3  cents  pel 
pound  to  market  it  ? 

12.  On  a  certain  date  hogs  sold  for  $9.75  per  hundred  pounds 
in  the  Chicago  market.  What  must  a  buyer  pay  for  hogs  in  the 
country  in  order  to  market  them  at  this  price  and  make  a  profit 
of  20  %,  making  no  allowance  for  cost  of  delivery  ? 

13.  At  what  market  price  must  1-year  4  %  bonds  be  offered  for 
sale,  in  order  that  the  buyer,  by  holding  them  until  maturity, 
may  make  8  %  on  his  investment? 

Suggestion.  —  The  profit  made  must  come  from  two  sources,  the  interest 
on  the  par  vakie,  which  is  $4,  and  the  excess  of  the  maturity  value  over  the 
price  paid  for  the  bonds.    If  x  is  the  price  paid,  100  —  x  +  4  =  .08  a;. 

14.  At  what  market  price  must  1-year  7  %  bonds  be  purchased, 
if  by  holding  them  until  maturity  the  purchaser  makes  10  %  on 
his  investment? 


THE  EQUATION  33 

15.  At  what  price  must  3-year  5  %  bonds  (bonds  that  mature 
in  3  years)  be  purchased,  if  by  holding  them  until  maturity  the 
purchaser  makes  6  %  on  his  investment  ? 

Suggestion.  —  If  x  is  the  cost  required,  the  annual  profit  due  to  excess  of 
maturity  value  over  cost  is ^^^. 

o 

16.  At  what  price  were  Lake  Shore  4's  bought  in  1910,  which, 
if  held  until  maturity  in  1931,  would  earn  the  buyer  4|  %  on  his 
investment  ? 

17.  Lackawanna  5's  bought  in  1910  and  maturing  in  1923,  if 
held  until  maturity,  would  pay  the  purchaser  6-f^  %  on  his  invest- 
ment.    What  was  the  price  of  Lackawanna  5's  when  bought  ? 

18.  Chicago  &  N.  W.  stock  bought  May  13,  1913,  and  sold 
May  13,  1913,  at  173,  after  paying  three  12  %  dividends,  would 
yield  the  purchaser  12J  %  on  his  investment.  At  what  was  the 
stock  bought  May  13,  1913? 

Suggestion.  —  If  n  =  amount  paid  for  a  $  100  share,  show  that 

115-=-^ +12  =  .  126  n. 
o 

19.  A  merchant  sold  an  article  at  a  gain  of  40%.  When  the 
cost  increases  $  90,  the  same  selling  price  will  be  a  loss  of  50  % , 
Find  the  cost. 

20.  A  weight  W  is  lifted  by  means  of  a 
rope  running  over  two  pulleys,  A  and  B. 
The  friction  in  each  pulley  increases  the  ten- 
sion in  the  rope  2%.  Find  ]r  if  the  force 
required  to  lift  it  is  400  pounds. 

*  This  method  of  distributing  the  loss  or  gain  at  maturity  does  not  consider 
interest  upon  the  annual  loss  or  gain.  For  a  full  discussion,  see  the  Stone' 
Millis  Secondary  Arithmetic,  page  90. 


CHAPTER  III 
POSITIVE  AND   NEGATIVE   NUMBERS 

22.  Extension  of  the  Number  System.  —  We  have  seen  in  Chap- 
ter  I  a  class  of  practical  problems  which  are  solved  by  use  of  the 
formulay  and  which  require  a  knowledge  of  literal  notation.  In 
Chapter  II  we  have  seen  another  class  of  problems  to  solve  which 
requires  a  knowledge  of  the  equation.  There  are  still  other  prob- 
lems, now  to  be  considered,  which  require  us  to  extend  our  idea 
of  number  to  include  a  new  hind  of  number. 

If  a  man  deposits  $  500  in  a  bank,  and  then  withdraws  $  300,  the  amount 
of  his  balance  is  $  500  —  $300,  or  $  200.  It  sometimes  happens  that  a  depos- 
itor withdraws  more  money  than  he  has  deposited^  or  makes  an  overdraft. 
For  example,  suppose  that  he  has  deposited  only  $250,  and  withdraws  $300. 
There  is  now  a  balance  of  $  50  against  him,  showing  that  he  owes  the  bank 
and  not  that  the  bank  owes  him.  The  balance  is  obtained  by  subtracting 
$300,  the  amount  withdrawn,  from  $250;  i.e.  the  balance  is  $250  — 


This  problem,  then,  requires  us  to  subtract  a  number  from  a 
smaller  number.  In  order  to  solve  such  problems,  our  idea  of 
number  must  be  extended  and  this  subtraction  made  possible. 

23.  Negative  Number.  —  The  remainder  resulting  from  subtract- 
ing a  number  from  a  smaller  number  is  called  a  negative  number. 
We  must  now  see  how  to  represent  a  negative  number. 

If  a  person  wishes  to  withdraw  $  100  from  a  bank,  he  may  withdraw  part 
at  a  time.  Thus,  he  may  withdraw  $75  with  one  check  and  then  $25  with 
another.  What  are  some  ways  that  he  may  withdraw  $250  from  a  bank, 
by  withdrawing  only  part  at  a  time  ? 

Similarly,  in  general,  one  number  may  be  subtracted  from 
another  by  separating  the  subtrahend  into  two  or  more  parts,  and 
subtracting  the  parts  one  at  a  time. 

34 


POSITIVE  AND  NEGATIVE  NUMBERS  35 

Hence,  in  attempting  to  subtract  7  from  4,  by  separating  7  into 

4  and  3,  we  get 

^  4-7  =  4-4-3  =  0-3. 

In  practice  the  zero  is  dropped,  and  the  remainder  0  —  3  written 
simply  —3. 

Similarly,       5-    9  =  5- 5  -  4  =  0 -4  or  -4. 
8- 14  =  8-8-6  =  0-6  or  -6. 

Hence,  a  negative  number  is  expressed  by  a  number  preceded  by 
the  minus  sign. 

It  is  clear  that  a  negative  number,  such  as  —  $  25,  indicates  a 
reserved  subtraction,  there  being  nothing,  when  it  stands  alone, 
from  which  to  subtract  it.  Thus,  in  $  100  -  $  150  =  $  100  -  $  100 
—  $50  =  —  $50,  the  remainder,  —  $50,  is  a  part  of  the  subtrahend 
that  is  not  subtracted  because  there  is  nothing  from  which  to 
subtract  it. 

Hence,  a  negative  number  is  in  nature  a  subtrahend. 

24.  Positive  Numbers. — For  the  sake  of  distinguishing  from 
negative  numbers,  the  arithmetical  numbers  with  which  we  have 
always  dealt  up  to  this  time  are  called  positive  numbers.  All 
numbers  with  which  we  are  familiar,  therefore,  are  to  be  classified 
as  either  positive  or  negative  numbers. 

25.  Use  of  Signs.  —  For  distinction  in  writing  pos-itive  and  nega- 
tive numbers,  the  positive  or  arithmetical  numbers  are  often  pre- 
ceded by  the  sign  +  when  standing  alone. 

Thus,  6  is  written  +6;  a  is  written  +  a. 

When  clearness  would  not  be  sacrificed,  however,  the  sign  -f 
may  be  omitted  from  a  positive  number.  If  no  sign  is  written 
before  a  number,  the  number  is  understood  to  be  positive. 

It  is  clear,  then,  that  the  signs  -h  and  —  have  two  uses:  to 
indicate  addition  or  subtraction,  and  to  indicate  the  positive  or 
negative  quality  or  character  of  numbers.  They  may  always  be 
considered  as  signs  of  addition  or  subtraction;  and  when  conven- 
ient, as  signs  of  quality. 


36  ELEMENTARY  ALGEBRA 

When  standing  alone,  the  positive  numbers  +1,  +2,  etc.,  are 
read  either  ''plus  1,"  "plus  2,"  etc.,  or  "positive  1,"  "positive  2," 
etc. ;  and  the  negative  numbers  —  1,  —  2,  etc.,  are  read  either 
"  minus  1,"  "  minus  2/'  etc.,  or  "  negative  1,"  "  negative  2,"  etc. 

EXERCISES 
Read  the  following  numbers : 

1.  5;  90;  -1;   -12;  x;   -  n;   \,   -  i;    -2. 

2  n  0 

What  negative  numbers  result  from  the  following  subtractions  ? 

2.  3-4.      3.    8-13.      4.    10-12.      5.    7-16.      6.   32-40. 
7.    $100 -$125.      8.    $500 -$650.      9.    $400  deposited  and 

$475  withdrawn.     10.    $250  deposited  and  $325  withdrawn. 

26.  Opposite  Numbers.  —  Since  a  negative  number,  such  as  —  5, 
always  implies  a  reserved  subtraction,  if  it  is  combined  with  a 
positive  number,  into  one  number,  it  tends  to  subtract  from,  or 
destroy  of,  the  positive  number  a  part  equal  to  itself. 

Thus,  when  6  and  —  4  are  combined  into  one  number,  —  4  destroys  4  of 
the  6,  leaving  2  as  the  result.  And  when  $10  and  —$15  are  combined, 
--  $  10  of  the  —  1 15  destroys  the  $  10,  leaving  —  $  5  as  the  result. 

Because  of  this  neutralizing  tendency  of  positive  and  negative 
numbers,  they  are  sometimes  called  opposite  numbers. 

If  two  numbers  such  as  +8  and— 8,  which  differ  only  in  their 
signs,  are  combined,  they  completely  neutralize  each  other.  Such 
numbers  are  said  to  have  the  same  absolute  value.  Thus,  8  is  the 
absolute  value  of  both  +  8  and  —  8. 

27.  Positive  and  Negative  Magnitudes.  —  As  seen  in  the  above 
illustrations,  deposits  and  withdrawals  of  money  in  a  bank  tend 
to  neutralize  each  other,  and  hence  are  positive  and  negative 
quantities,  respectively.  There  are  many  other  concrete  quanti- 
ties or  magnitudes  which  are  capable  of  existing  in  two  opposite 
states  or  senses  such  that  one  tends  to  neutralize  or  destroy  an 


POSITIVE  AND  NEGATrVE   NUMBERS  37 

equal  amount  of  the  other.  Consequently,  their  numerical  values 
are  represented  by  positive  and  negative  numbers.  Several  of 
these  are  illustrated  in  the  following  exercises. 

EXERCISES 

1.  If  in  one  of  two  sales  of  goods  I  lose  $10,  and  in  the  other 

I  make  a  profit  of  $  15,  what  is  the  net  result  ? 

Loss  and  profit  are  thus  neutralizing  quantities,  each  tending 
to  destroy  an  equal  part  of  the  other. 

If  profit  is  called  positive,  what  will  loss  be  called  ? 

If  $15  profit  is  written  +15,  what  will  express  $10  loss? 

2.  Express  with  +  and  —  signs  the  following:  $25  loss; 
$40  profit;  $15  loss;  $6  loss;  $20  profit. 

3.  If  I  had  $20  in  my  pocket,  then  spent  $25  for  a  suit  of 
clothes,  I  would  have  left  $20-  $25,  or  -  $5.  What  does  the 
—  $  5  in  this  case  mean  ? 

4.  Express  with  +  and  —  signs  the  following:  $50  credit; 
$18  debt;  $25  debt;  $60  cash  in  hand;  $100  borrowed;  $20 
loaned ;  $  15  spent. 

5.  The  number  of  people  entering  a  store  during  a  certain  hour 
was  216,  and  the  number  coming  out  was  250.  Express  the  rela- 
tion of  these  numbers  by  +  and  —  signs.  Also  express  the 
excess  of  the  number  coming  out  over  the  number  entering. 

6.  The  amount  of  water  pumped  into  a  tank  during  an  hour 
was  1900  gallons;  and  during  the  same  hour  three  locomotives 
withdrew,  respectively,  675  gallons,  750  gallons,  and  650  gallons. 
Express  with  +  and  —  signs  these  quantities,  and  also  the  excess 
withdrawn  during  the  hour. 

7.  A  locomotive  pulls  a  train  with  a  force  of  35  tons.  The 
resistance  of  the  train  is  20  tons.  If  the  pull  of  the  locomotive  is 
positive,  represent  by  +  and  —  signs  the  two  forces. 

8.  A  balloon  lifts  up  with  a  force  of  200  pounds.  How  much 
weight  must  be  attached  to  keep  the  balloon  from  rising?  If  the 
lifting  force  is  -|-,  what  represents  the  weight? 


38  ELEMENTARY  ALGEBRA 

9.  A  balloon  lifts  with  a  force  of  250  pounds,  and  a  weight  of 
175  pounds  is  attached  to  it.  If  the  weight  is  called  positive, 
what  will  represent  the  lifting  force  ?  What  will  represent  the 
result  when  the  weight  is  attached  ? 

10.  A  battleship  has  a  displacement  of  20,000  tons  (sinks  into 
the  water  until  it  displaces  20,000  tons  of  water).  Express  with 
+  and  —  signs  the  weight  of  the  ship  and  the  upward  pressure 
of  the  water. 

11.  A  diver  who  weighs  165  pounds  displaces  156  pounds  of 
water  (is  buoyed  up  with  a  force  of  156  pounds).  Kepresent  by 
-f-  and  —  signs  these  forces  and  the  resultant  force  with  which  he 
tends  to  sink  to  the  bottom. 

12.  If  the  thermometer  is  5°  above  zero  at  noon,  and 
falls  15°  by  night,  what  is  the  temperature  by  night? 

1 212*     5°— 15°  =  what  ?     How   else   may   10°    below   zero    be 
written  ?     How  would  temperature  above  zero  be  written  ? 

13.  On  Jan.    6,  1910,  the  Weather  Bureau  reported 
the   temperature  at  several  different  places  as   follows 
Chicago,    -2°;  Davenport,    -18°;  Des  Moines,   -12° 
Kansas  City,  —  2° ;  Madison,   —  14° ;  Milwaukee,  —  8° 
Peoria,  -6°;    Pueblo,    -6°;    St.    Paul,    -16°.     Were 
these  temperatures  above  or  below  zero  ?    What  would 
they  be  if  the  signs  were  +  ? 

14.  Certain  railroad  stock  that  sold  at  7  above  par  in 
January  fell  to  3  below  par  in  June.  Since  the  value  of 
the  stock  was  reduced  10  points,  how  may  the  3  below 
par  be  represented  algebraically  ? 

15.  Eepresent  by  -h  and  —  signs  the  following  prices  of  stock 
which  show  the  change  in  6  months :  January,  10  below  par ; 
February,  6  below  par ;  March,  1  below  par ;  April,  5  above  par ; 
May,  8  above  par ;  June,  2  above  par. 

16.  Augustus  Caesar  was  ruler  of  the  Roman  Empire  from  —  31 
to  4- 14.     Express  these  dates  in  other  terms. 


POSITIVE  AND  NEGATIVE   NUMBERS  39 

17.  Arcliiniedes,  the  greatest  mathematician  of  antiquity,  was 
born  about  the  year  —287,  and  was  slain  by  a  Koman  soldier 
while  studying  a  geometrical  figure  that  he  had  drawn  in  the  sand 
in  —  212.     Express  these  dates  in  other  terms. 

18.  If  a  file  of  soldiers  moves  10  paces  to  the  front,  then  6  paces 
to  the  rear,  it  advances  how  many  paces  from  its  first  position  ? 
If  it  moves  6  paces  to  the  front,  then  10  paces  to  the  rear,  it  ad- 
vances 6  — 10,  or  how  many  paces  ?  If  motion  forward  is  positive, 
what  is  motion  backward  ? 

19.  A  switch  engine,  in  making  up  a  train,  moves  forward 
100  yards,  then  backward  150  yards.  The  two  are  equivalent  to 
what  single  motion  ?  Represent  by  appropriate  signs  these  two 
motions  and  the  one  equivalent  motion. 

20.  If  longitude  west  is  positive,  longitude  east  negative,  lati- 
tude north  positive,  and  latitude  south  negative,  in  what  country 
is  a  city  whose  longitude  is  +  106°  and  latitude  -f  40°?  Longitude 
-16°  and  latitude  -f  48?  Longitude  -40°  and  latitude  -  15°? 
Consult  a  map. 

21.  The  longitude  of  the  city  of  St.  Petersburg,  Russia,  is  given 
as  —  31°.  This  means  that  it  is  31°  in  what  direction  from  the 
Prime  Meridian?  Washington,  D.C.,  is  in  77°  west  longitude. 
How  else  may  this  be  expressed  ? 

22.  The  latitude  of  New  York  City  is  about  -f-  41°.  What  will 
express  the  latitude  of  Cape  Town,  Africa,  which  is  about  34°  south 
of  the  equator  ? 

28.  Numbers  Represented  by  Distances.  —  In  the  preceding  ex- 
ercises, it  has  been  seen  that  many  magnitudes  may  be  represented 
numerically  by  positive  and  negative  numbers.  These  include 
bank  deposits  and  withdrawals,  loss  and  gain  in  business,  income 
and  expense,  credits  and  debts,  increases  and  decreases  of  physical 
quantities,  temperature  above  and  below  zero,  dates  in  history 
before  and  after  the  birth  of  Christ,  stocks  above  and  below  par, 
forces  acting  in  opposite  directions,  motions  in  opposite  directions, 


40  ELEMENTARY  ALGEBRA 

and  longitude  east  and  west  and  latitude  north  and  south.     Many 
others  might  be  suggested. 

Since  motions  in  opposite  directions  tend  to  neutralize  each 
other,  the  distances  moved  through  in  opposite  directions  may 
themselves  be  considered  as  positive  and  negative.  In  represent' 
ing  distances  as  positive  and  negative,  the  signs  serve  to  tell  the 
directions  in  which  the  distances  are  measured,  as  in  the  case  of 
longitude  and  latitude.  It  follows  that  positive  and  negative 
numbers  may  be  represented  to  the  eye  by  distances  measured  on 


-9-8 -7-6-5-4 -3 -2-1    0    1    2   3  4  5    6   r  8   9 
I     I     I     t     I     I     I     I     I  .,  I I—  J  li.  I.    I    1 1,  1 1     I — I     I  , 


A 

a  line.  Positive  numbers  are  represented  by  distances  measured 
to  the  right  from  a  starting  point  A,  and  negative  numbers  by  dis- 
tances measured  to  the  left, 

EXERCISES 

1.  On  the  above  line,  how  will  —  12  be  represented?  +10? 
-21?  +31?  -40?  +100? 

2.  Draw  a  horizontal  line  and  mark  the  middle  point  O.  Then 
mark  on  it  distances  representing  the  following  numbers,  a  unit 
being  represented  by  a  quarter  of  an  inch  :  +  2,  +  5,  +  8,  +  10, 
+  61,  -3,    -4,  -7,  -12,  -41,  -lOi 

3.  On  a  horizontal  line  represent  the  following  dates  (i.e.  years) : 
—  31,  beginning  of  Augustus  Caesar's  reign ;  +14,  end  of  Au- 
gustus Caesar's  reign ;  —  146,  fall  of  the  Eoman  Empire ;  +  70, 
destruction  of  Jerusalem  by  Titus ;  —  275,  death  of  Euclid,  who 
wrote  the  first  great  book  on  geometry ;  —  287,  birth  of  Archi- 
medes, the  greatest  mathematician  of  antiquity. 

4.  On  a  vertical  line,  represent  the  following  temperatures: 
+  5°;  +10°;  +15°;  +20°;  -5°;  -10°;  -15°;  -20°. 

5.  The  minimum  temperatures  at  St.  Paul  on  the  six  days  be- 
ginning Eeb.  9, 1910,  were  +  le**,  -  10°,  +  6°,  +  4°,  +  22°.  Eep- 
resent  these  on  a  line. 


POSITIVE  AND  NEGATIVE  NUMBERS 


41 


6.  The  longitude  of  New  York  is  +  74°,  of  Berlin  - 13°,  of 
Pekin  -  116°,  of  Calcutta  -  88°,  of  Pittsburg  +  80°,  of  St.  Louis 
-f  90°,  of  Rome  —12°.  Draw  a  horizontal  line  and  mark  off  on  it 
the  longitudes  of  these  places. 

29.  Graphs.  —  The  following  table  gives  the  minimum  temper- 
ature at  Chicago  on  each  of  the  seven  consecutive  days  beginning 
with  Jan.  2,  1910; 


Date 

Jan.  2 

Jan.  8 

Jan.  4 

Jan.  5 

Jan.  6 

Jan.  7 

Jan.  8 

Temp. 

21° 

1° 

1° 

4° 

-4° 

-6° 

14° 

Suppose  that  a  sheet  of  paper  is  ruled  by  horizontal  and  verti- 
cal lines,  as  shown  in  the  following  diagram,  each  vertical  line 
representing  the  thermometer  scale.  By  marking  the  temper- 
ature for  each  date  on  the  corresponding  vertical  line,  and  joining 
the  consecutive  points  thus  marked  by  straight  lines,  the  heavy 
broken  line  is  obtained,  showing  to  the  eye  the  variation  from 
day  to  day. 

25 


Jan.  2      Jan.3      Jan.4       Jan.5      Jan.6      Jan.7       Jcxn.8 


Such  diagrams  as  this  are  called  graphs.  They  are  much  used 
in  practical  work  to  depict  to  the  eye  graphically  the  changes  in 
quantities,  such  as  temperature,  death  rates,  population,  imports 
and  exports,  cost  of  living,  etc. 

Note. — Ruled  paper,  often  called  ''squared  paper"  or  "coordinate 
paper,"  may  be  obtained  cheaply  for  drawing  graphs.  It  is  iniled  accu- 
rately into  small  squares,  and  saves  much  time  when  used.  This  paper  will 
be  needed  in  other  exercises  later. 


42 


ELEMENTARY  ALGEBRA 


EXERCISES 

1.  The  following  table  gives  the  minimum  temperature  at 
St.  Paul,  Minn.,  on  each  of  seven  consecutive  days,  beginning 
Jan.  5,  1910 : 


Date 

Jan.  5 

Jan.  6 

Jan.  7 

Jan.  8 

Jan.  9 

Jan.  10 

Jan.  11 

Temp. 

6° 

-16° 

-12° 

-4^ 

-12° 

2° 

20° 

Draw  a  graph  showing  the  variation  in  temperature  during  the 
period. 

2.  According  to  the  report  of  the  Weather  Bureau,  on  Jan.  5, 
1910,  the  height  of  the  Ohio  River  above  low-water  mark  at 
Pittsburg  was  11.3  feet.  The  amounts  in  feet  by  which  it  rose 
or  fell  on  the  seven  consecutive  days  thereafter  were  as  follows, 
the  negative   sign   indicating   fall :    —  2.1,  —  2.8,  1.2,  1.3,  —  2,2, 

-  2.7,  -  1.4. 

Draw  a  graph  showing  the  rise  and  fall  of  the  river  from  day 
to  day.  Measure  the  heights  of  the  river  on  vertical  lines,  mark- 
ing the  lowest  horizontal  line  low-water  mark. 

3.  The  latitude  of  a  ship  going  east  at  noon  on  each  of  six  con- 
secutive days  was  as  follows :    —  1°,  equator,   -|-  1°,  -f  3°,   +  1°, 

—  1°.     Draw  a  graph  showing  the  course  of  the  ship. 

4.  The  highest  points  reached  by  certain  railroad  stock  during 
the  12  months  of  a  year  were  as  follows :  10  above  par,  6  above 
par,  2  below  par,  5  below  par,  par,  2  below  par,  3  above  par,  7 
above  par,  10  above  par,  15  above  par,  18  above  par,  12  above  par. 

Draw  a  graph  showing  the  variation  in  price  of  the  stock  dur- 
ing the  year. 

30.  Addition  of  Positive  and  Negative  Numbers.  —  The  addition 
of  positive  and  negative  numbers  may  be  indicated  by  writing 
them  in  succession  with  their  signs. 

Thus,  to  indicate  the  addition  of  +  6,  +  2,  —  4,  -f-  3,  and  —  5,  we  write 
-f-6  +  2  —  4  +  3  —  5.  The  first  term  being  positive,  we  may  omit  its  sign, 
giving  6  +  2-4  +  3-6. 


POSITIVE  AND   NEGATIVE  NUMBERS  43 

Positive  and  negative  numbers  to  be  added  may  be  placed  _  „ 

also  in  columns,  with  the  signs  attached,  as  in  the  margin.  « 

Since  positive  numbers  are  in  nature  ordinary  arithmet-  /, 

leal  numbers,  their  sum  is  obtained  by  adding  their  absolute 

values,  and  is  positive. 

Thus,  a  bank  deposit  of  .$  125  and  another  of  $  100  together  make  a  de- 
posit of  $225.     The  sum  of  +  4  and  -f  8  is  4- 12. 

Also,  to  subtract  each  of  two  numbers  in  succession  is  equiva- 
lent to  subtracting  their  sum.  Hence,  two  negative  numbers, 
since  each  is  in  nature  a  subtracted  number,  may  be  combined  by 
adding  their  absolute  values,  and  the  sum  is  negative. 

Thus,  a  withdrawal  from  the  bank  of  $  25  and  an  additional  withdrawal  of 
$75  together  make  a  withdrawal  of  $  100.    The  sum  of  —6  and  —10  is  —16. 

Therefore, 

(1)  To  add  two  numbers  with  like  signs,  find  the  sum  of  their 
absolute  values,  and  prefix  the  sign  common. 

In  adding  two  numbers  with  unlike  signs,  the  one  with  the 
greater  absolute  value  may  be  separated  into  two  parts,  one  of 
which  just  neutralizes  the  number  with  the  smaller  absolute 
value,  leaving  as  result  the  other  part  of  the  numerically  greater 
number. 

Thus,  to  add  +  7  and  —  3,  by  separating  +  7  into  +  3  and  +  4,  we  have 

-1-3  +  4  or  4-  7 

-3  -3 

0  +  4  +4 

And,  to  add  —  9  and  +  4,  by  separating  —  9  into  —  4  and  —  6,  we  have 

—  4  —  5  or  —  ^ 

+  4  +4 

0-5  =.  8  J 

Hence,  the  following  rule  is  evident : 

(2)  To  add  two  numbers  with  unlike  signs,  find  the  difference  of 
their  absolute  values,  and,  prefix  the  sign  of  the  one  ivith  the  greater 
absolute  value. 


44  ELEMENTARY  ALGEBRA 

Evidently  three  or  more  positive  and  negative  numbers  may  be 
added  by  adding  two  at  a  time,  taking  them  in  the  order  written. 
In  the  steps  of  the  addition  use  rules  (1)  and  (2). 

A  second  method  sometimes  used  when  there  are  both  positive 

and  negative  numbers  to  be  added  is  to  add  the  positive  and  the 

negative  numbers  separately,  and  then  combine  these 

two  sums.  ~-  ^ 

+  8 
Thus,  in  this  column,  adding  upwards,  the  partial  sums  are  _  g 

—  2,  4-  6  ,  and  +  3,  the  result.     Or,  the  sum  of  —  3  and  —  6  is  ^_  4 

—  9,  and  that  of  +  8  and  +  4  is  +12,  and  the  sum  of  these  Vs 
sums  is  +  3. 

Note. — If  desirable,  the  numbers  in  the  preceding  illustration  — 

might  be  rewritten  in  the  order  shown  in  the  margin.     This  fun-  — 

damental  principle  that  numbers  to  be  added  may  be  arranged  in  +  8 ) 

any  order  is  known  as  the  Law  of  Order  in  Addition.  -f-  4  j 

The  principle  that  numbers  to  be  added  may  be  grouped  in  any 
manner  —  by  which  we  find  the  sum  of  the  —  3  and  the  —  6  and  of  the  +  8 
and  +  4  and  then  the  sum  of  these  sums,  as  here  indicated — is  known  as  the 
Law  of  Grouping  in  Addition. 

EXERCISES 

Find  the  sums  of  the  following : 

1.  A  profit  of  $12  and  a  profit  of  $16. 

2.  A  loss  of  $10  and  a  loss  of  $23. 

3.  A  profit  of  $  15  and  a  loss  of  $  6. 

4.  A  profit  of  $  18  and  a  loss  of  $  30. 

5.  A  rise  in  temperature  of  14°  and  a  rise  of  6°. 

6.  A  fall  in  temperature  of  8°  and  a  fall  of  17°. 

7.  A  rise  in  temperature  of  7°  and  a  fall  of  16°. 

8.  A  credit  of  $  250  and  a  debt  of  $  325. 

9.  A  debt  of  $  136  and  a  debt  of  $  224. 

10.    +6  -8  +9  -12  -20  +8 

_2  -7  +5  +17  -&^  -16 


POSITIVE  AND   NEGATIVE  NUMBERS  45 


11. 

-25 
+  32 

- 12         +13 

- 17         +17 

-    1 

+  14 

+    1         -98 
-  30         +16 

12. 

-14 
+  21 

+    7         -    8 
-18         +7 

+    3 
-12 

- 14         +20 
-3         +16 

13. 

-  $12         +$27         -    7  1b.         + 

-  $   6         -$36         +12  lb.         + 

-  8°  longitude         +    6°  latitude 

-  17°  longitude         -  14°  latitude 

13  lb.         +  14  mi. 
19  lb.         -  18  mi. 

14. 

25  ft.  forward 
36  ft.  backward 

15. 

-2    16. 
+  6 
+  3 
-8 
-4 

+  12     17.   -   4 

-  7            -   9 

-  3            -   3 
+   6           -    1 
+   5            +12 

18.    +    6 

-  3 

-  4 

+  12 
+   7 

19.   —   5     20.   +    8 
+   8            -16 
+  14            -21 
-   3            -   7 
+   6            -   1 

21. 

-3.25 
+  4.09 

- 12.08 
- 10.37 

+   .62 
-6.07 

-  4.125 
+  1.450 

22. 

-^ 

+  6f            -   i            +1| 
_4|             -51             +   J 

23.  5,  -.  8,  7,  -  4,  -  12,  3. 

24.  -9,-6,-2,  10,  -  8,  9,  - 1. 

25.  16,.  5,  9,  -  5,  3,  -  7,  -  2,  6. 

26.  The  Ahraes  papyrus,  the  earliest  mathematical  book  of 
which  there  is  any  record,  was  written  about  —  1700.  The  next 
great  mathematical  book  was  Euclid's  Elements,  written  about 
1400  years  later.  What  was  the  date  that  Euclid  wrote  the 
Elements? 

27.  Euclid  was  born  about  —330.  Biophantus,  who  wrote 
one  of  the  first  books  on  algebra,  was  born  750  years  later.  At 
what  date  was  Diophantus  born  ? 


46  ELEMENTARY  ALGEBRA 

28.  A  freight  car  is  running  at  the  rate  of  20  feet  per  second. 
A  brakeman  walks  to  the  rear  on  the  top  of  the  car  at  the  rate 
of  5  feet  per  second.  Express  by  +  and  —  signs  these  rates 
and  the  rate  and  direction  of  his  motion  with  reference  to  the 
ground. 

29.  If  the  basket  of  a  balloon  weighs  280  pounds,  the  instru- 
ments 57  pounds,  the  sandbags  800  pounds,  and  the  balloon  itself 
—  1400  pounds,  what  is  the  total  weight  of  the  balloon  and 
contents  ? 

30.  A  boat  using  both  sails  and  steam  is  driven  against  a  cur- 
rent by  its  steam  power  at  the  rate  of  10  miles  per  hour,  and  by 
the  wind  at  the  rate  of  3  miles  per  hour.  The  current  flows  at 
the  rate  of  4  miles  per  hour.  Express  these  rates  by  +  and  — 
signs,  and  indicate  their  sum.    What  is  the  boat's  rate  of  progress  ? 

31.  A  work  train  travels  12  miles  north,  then  35  miles  south, 
then  8  miles  north.  By  use  of  -f-  and  —  signs  express  these  dis- 
tances, their  sum,  and  hence  its  final  distance  and  direction  from 
the  starting  point. 

32.  The  temperature  falls  15°,  rises  18°,  falls  10°,  then  rises  6°. 
By  use  of  -{-  and  —  signs  express  these  changes  and  their  net 
result. 

33.  The  rise  and  fall  in  feet  of  the  Mississippi  River  at  Yicks- 
burg  on  each  of  the  seven  days  beginning  Jan.  6,  1910,  were 
given  by  the  Weather  Bureau  as  follows:  —  .2,  —  .2,  —  .4,  —  .2, 
-f  .4,  -h  .2,  +  .4.  The  sign  —  indicated  fall.  Find  the  net  change 
during  that  period. 

34.  The  average  of  two  or  more  numbers  is  their  sum  divided 
by  the  number  of  them. 

Find  the  average  of  the  following  temperatures:  8  a.m.,  —7°; 
9  A.M.,  -  6° ;  10  A.M.,  -  3° ;  11  a.m.,  -  1° ;  12  Noon,  + 1° ;  1  p-m.^ 
-f-4°;  2  P.M.,  -f-7°;  3  p.m.,  4-8°;  4  p.m.,  +8°;  5  p.m.,  -f  5°, 
6  p.m.,  +3°;  7  P.M.,  -f  1°. 


POSITIVE  AND  NEGATIVE  NUMBERS  47 

35.  At  Madison,  Wis.,  the  minimum  temperatures  on  the  seven 
days  beginning  Jan.  4,  1910,  were  as  follows:  —12°;  +10°; 
-14°;  -18°;  +6°;  -6°;  -10°.  Find  the  average  tempera- 
ture for  the  week. 

36.  At  Cheyenne,  Wyo.,  the  temperatures  for  six  days,  begin- 
ning Jan.  3,  1910,  were  as  follows:  -10°;  -2°;  0°;  —2°; 
+ 14° ;   + 10°.     Find  the  average  temperature  for  the  six  days. 

37.  If  one  place  is  midway  between  two  other  places,  its  lati- 
tude or  longitude  equals  one-half  the  sum  of  their  latitudes  or 
longitudes. 

The  latitude  of  Panama  is  approximately  +  8°,  and  of  Chicago 
+  42°.  Find  the  latitude  of  Key  West,  Florida,  which  is  approxi- 
mately midway  between  these  two. 

38.  The  longitude  of  New  Orleans  is  -f  90°,  and  of  Pekin, 
China,  —  116°.  Berlin,  Germany,  is  approximately  midway  be- 
tween these  two  places.     What  is  its  longitude  ? 

31.  Subtraction  of  Positive  and  Negative  Numbers.  —  To  subtract 
a  number  (subtrahend),  is  to  find  a  number  (remainder)  which 
added  to  it  gives  the  minuend. 

Thus,  to  subtract  6  from  9  is  to  find  a  number  (3)  which  added  to  6  gives  9. 

Hence,  the  rule  for  subtracting  positive  and  negative  numbers 
may  be  obtained  from  addition. 

(1)  What  number  added  to  +  3  gives  +  6  ? 

+  5  +5 

Hence,  by  subtracting,    +  3      But,  by  adding,   —  3 

+  2  +1 

Ilence,  to  subtract  +  3  from  +  6  is  equivalent  to  adding  —  3  to  +  6. 

(2)  What  number  added  to  +  3  gives  —  5  ? 

-6  -5 

Hence,  by  subtracting,    +  3      But,  by  adding,    —  3 

Hence,  to  subtract  +  3  from  -  5  is  equivalent  to  adding  —  3  to  —  6. 


48  ELEMENTARY  ALGEBRA 

(3)  What  number  added  to  —  3  gives  +  5  ? 

Show,  as  above,  that  to  subtract  -^  3  from  +  5  is  equivalent  to  adding 
-I-  3  to  +  5. 

(4)  What  number  added  to  —  3  gives  —  5  ? 

Show,  as  above,  that  to  subtract  —  3  from  —  5  is  equivalent  to  adding 
+  3  to  -  5. 

From  these  examples  it  is  evident  that  to  subtract  one  number 
(subtrahend.)  from  another  (minuend)  is  equivalent  to  adding  the 
subtrahend,  with  its  sign  changed,  to  the  minuend.     Therefore, 

To  suhtr act,  first  change  the  sign  of  the  subtrahend,  then  proceed 

as  in  addition. 

—  4  —  4 

Thus,  to  subtract    —  7   change  to    +  7   then  add. 

Note.  —  The  pupil  should  learn  not  to  rewrite  the  problem  with  the  sign 
changed  in  the  subtrahend,  but  to  make  the  change  mentally,  and  add  at 
once. 

EXERCISES 

1.  What  number  added  to  —  6  gives  —  4  ? 

2.  What  number  added  to  8  gives  —  2  ? 

3.  What  number  added  to  —  4  gives  6  ? 

4.  What  number  added  to  12  gives  7  ? 

5.  What  number  added  to  —  15  gives  —  8  ? 

6.  What  number  added  to  —  5  gives  0  ? 

7.  What  added  to  $  7  loss  will  give  $  9  gain  ? 

8.  AVhat  added  to  a  withdrav^^al  of  $  125  from  a  bank  will 
give  a  balance  of  $  150  ? 

9.  What  added  to  a  rise  of  12°  temperature  vrill  give  a  net 
fall  of  18°? 

10.  What  added  to  a  fall  of  15°  temperature  will  give  a  net 
fall  of  8°  ? 

Subtract  the  following : 

11.  -h    7        +12        +2        +20        +9        +16  0 
-6        -3        -10        -8        -14        -4        -6 


POSITIVE  AND  NEGATIVE  NUMBERS 


49 


12.    -    3 

+   5 

-    7 
+   6 

-25 

+  15 

-14 

+  16 

-   9 
+   6 

-  7 
+   8 

0 

+  15 

13.    +12 
+   3 

+   9 
+   5 

+   7 
+   9 

+  15 

+  20 

+  21 
+  16 

+  12 
+  20 

+  40 

+  55 

14.    -    9 
-    6 

-  8 

-  4 

-  3 

-  5 

-12    - 
-15 

-  9 

-  4 

-  7 
-10 

-16 
-17 

15.    —    3 

+    7 

+   6 
-15 

-  5 

-  7 

+  10 
+   6 

-16 
+   4 

+  5 
+  12 

+    9 
-    6 

16.  What  is  indicated  by  —  3  —  (—  2)  ?     Find  the  value. 

17.  What  is  indicated  by  +  8  —  (+  12)  ?     Find  the  value. 

18.  Find  the  value  of  the  following :  5 -(-5);  -3 -(-10); 
6-(+8);  -9-(-12);  6-(+12);  -7-(+8);  9-(-6). 

19.  Subtract :    -6.25     +  .23     -  .2  ft.     +3.5  pk.     -12.3  lb. 

+3.85     -4.17     -1.6  ft.     +7.4  pk.     -15.61b. 


20.    Subtract : 


+2 


+5f       +  f 
-2i        -i\ 


-16J 
+  6J 


21.  A  thermometer  registered  one  day  — 15°,  and  the  next  day 

—  9°.     What  was  the  change  in  temperature  ? 

22.  A  thermometer  registered  one  day  +  10°,  and  the  next  day 

—  7°.     What  was  the  change  in  temperature  ? 

23.  Give   the  range  of  temperature  in  the  following  places 
(Jan.  1,  1908  to  Jan.  1,  1909) : 


Place 

Extremes 

Place 

Extremes 

Montgomery,  Ala.  .     . 
Little  Rock,  Ark.     .     . 

Denver,  Col 

Washington,  D.C.    .     . 

Bois6,  Ida 

Des  Moines,  la.  .     .     . 

107 
106 
106 
104 
111 
109 

-    5 
-12 
-29 
-15 
-28 
-30 

Louisville,  Ky.  .     .     . 
Boston,  Mass.    .     .     . 
Duluth,  Minn.  .     .     . 
Havre,  Mont.     .     .     . 
New  York,  N.Y.    .     . 
Chicago,  111.       .     .     . 

107 
102 
99 
108 
100 
103 

-20 
-13 
-41 
-65 
-    6 
-23 

50  ELEMENTARY  ALGEBRA 

24.  What  is  the  difference  in  time  between  the  year  4-  36  and 
the  year  —  40  ? 

25.  Augustus  Caesar  lived  from  —60  to  +14.  How  old  was 
he  when  he  died  ? 

26.  Thales,  the  founder  of  the  first  Greek  school  of  mathe- 
matics and  philosophy,  was  born  in  the  year  —  640  and  died  in 
the  year  —  550.     At  what  age  did  he  die  ? 

27.  Pythagoras,  who  is  believed  to  have  given  the  first  proof 
that  in  a  right  triangle  the  square  on  the  hypotenuse  equals  the 
sum  of  the  squares  on  the  other  two  sides,  was  born  in  —  569  and 
murdered  in  —  500.     At  what  age  was  he  murdered  ? 

28.  Euclid  wrote  his  geometry,  called  the  Elements,  in  about 
—  300.  By  + 1300  the  book  had  been  carried  into  western 
Europe  and  translated  there.  How  long  was  this  after  the  book 
had  been  written  ? 

29.  What  is  the  difference  in  longitude  between  Chicago  and 
Berlin,  the  longitude  of  the  former  being  approximately  +  88° 
and  of  the  later  approximately  —  13°  ? 

30.  During  the  month  of  December,  1909,  the  maximum  tem- 
perature at  Chicago,  as  recorded  by  the  Weather  Bureau,  was 
-f  55°,  and  the  minimum  —  7°.  What  was  the  amount  of  variar 
tion  during  the  month  ? 

32.  Multiplication  of  Positive  and  Negative  Numbers.  —  Multi- 
plication as  defined  in  arithmetic  is  a  short  form  of  addition, 
where  the  addends  are  all  equal.  Eor  example,  3  X  $  25  means 
$  25  +  $  25  +  $  25.  This  definition  is  only  a  special  case  of  a 
more  general  definition  which  may  be  stated  in  the  following  way: 

To  multiply  one  number  (multiplicand)  by  a  second  number  (mul- 
tiplier) is  to  use  the  first  as  we  use  1  (unity)  to  obtain  the  second. 

Thus,  in  multiplying  3  by  5,  to  obtain  5  by  using  1,  we  take 
1  +  1  +  1  +  1  +  1  =  5. 

Hence,  to  obtain  6  x  3,  we  take 

34-3  +  3  +  3  +  3  =  16. 


POSITIVE  AND  NEGATIVE  NUMBERS  51 

From  this  definition  are  obtained  the  rules  for  multiplying 
positive  and  negative  numbers. 

(1)  To  obtain  the  multiplier  +  3  we  must  add  three  I's  j  i.e, 

+  3  =  1  +  1  +  1. 
Hence,  to  obtain  (+  3)(+  5)  we  must  add  three  +  5's. 
That  is,  (+  3)(+  5)  =  +  5  +  5  +  5  =  +  15,  the  product. 

And  to  obtain  (+  3)(—  5)  we  must  add  three  —  5's. 
That  is,  (+  3) ( -  5)  =  -  5  —  5  -  5  =  —  16,  the  product. 

(2)  To  obtain  the  multiplier  —  3  we  must  subtract  three  I's  from  0  ;  i.e. 

_3  =  0-l-l-l. 

Hence,  to  obtain  (— 3)(+  5)  we  must  subtract  three  +  5's  from  0;  i.e. 
change  their  signs  and  add  them. 

Hence,  (—  3)(+  6)  =  —  6  —  6  -  5  =  —  15,  the  product. 

And  to  obtain  (  —  3)  (—  5)  we  must  subtract  three  —  6's  from  0  ;  i.e.  change 
their  signs  and  add  them. 

Hence,  (—  3)  (—  5)  =  +  5  +  5  +  5  =  +  15,  the  product. 

From  these  examples  the  following  principles  are  evident : 

(1)  The  product  of  two  numbers  with  like  signs  is  positive. 

(2)  Tlie  product  of  two  numbers  with  unlike  signs  is  negative. 

(3)  Tlie  absolute  value  of  the  product  of  two  numbers  is  the  prod* 
net  of  the  absolute  values  of  the  numbers. 

In  symbols :  (+  a)(+  &)  =  +  a6. 

(— a)(-&)= +a6. 
(+a)(-6)=-a6. 
(-a)(+6)  =  -a6. 

EXERCISES 
Give  orally  the  products  of  the  following : 

1.  +3,  +7.                  5.    -2,-10.  9.  +5,-7. 

2.  -6,-2.                   6.    +7,  +-3.  10.  +8,-3. 

3.  +4,  +5.                  7.    -2,  +5.  11.  -2,-15. 

4.  -  8,  -  3.                   8.    -  3,  +  6.  12.  -  7,  +  4. 


52  ELEMENTARY  ALGEBRA 


13. 

+  6,-8. 

19. 

-  5,  - 15. 

25. 

-i-f 

14. 

-5,  -12. 

20. 

+  6,  +20. 

26. 

+  H, -|. 

15. 

+  12,  +4, 

21. 

-10,  +14 

27. 

-  2.5,  -  2.5. 

16. 

-9,-8. 

22. 

-h-h 

28. 

+  1.6,  -  1.6. 

17. 

+  12,-8. 

23. 

+  1,  -|. 

29. 

-.8,  +.7. 

18. 

-7,  +12. 

24. 

-i+|. 

30. 

-1.3,  -1.1. 

31.  If  the  lifting  force  of  a  balloon  is  +,  the  weight  of  the 
ballast  is  — .  Adding  weights  is  indicated  by  +,  and  removing 
weights  by  — .  When  5  weights,  each  of  20  ponnds,  are  thrown 
overboard,  what  is  the  effect  on  the  upward  pull  of  the  balloon  ? 
Show  that  this  is  equivalent  to  saying  that  (—  5)(—  20)  =  +  100. 

32.  When  four  weights,  each  of  15  pounds,  are  added  to  a 
balloon,  what  is  the  effect  on  the  upward  pull  of  the  balloon? 
Show  that  this  is  equivalent  to  saying  that  (+  4)(—  15)  =  —  60. 

33.  If  10  boxes  of  goods,  each  weighing  250  pounds,  are  removed 
from  a  freight  boat,  what  is  the  effect  of  the  downward  pressure 
of  the  boat  in  the  water  ?  Show  that  this  is  equivalent  to  saying 
that  (- 10) (+  250)  =  -  2500. 

33.  The  Product  of  More  than  Two  Factors.  —  The  product  of 
more  than  two  factors  is  obtained  by  performing  one  multiplication 
at  a  time,  according  to  the  principles  in  §  32. 

Thus,  to  find  (-  2)(-  3)(+  4)(-  5),  we  have 
(-2)C-3)  =  +6 
(-H6)(+4)  =+24 
(+  24)(-  5)  =  -  120,  the  product. 

Show  that  the  following  rule  holds  for  finding  the  product  of 
more  than  two  factors : 

Find  the  product  of  the  factors  regardless  of  signs,  and  prefix  -\-  or 
—  according  as  the  number  of  negative  factors  is  even  or  odd. 

34.  Powers  of  Positive  and  Negative  Numbers.  —  If  the  factors 
are  all  equal,  what  is  the  product  called  ? 


POSITIVE  AND  NEGATIVE  NUMBERS  53 

The  rule  in  §  33  will  serve  for  finding  powers  of  positive  and 
negative  numbers. 

Thus,  (+  5)2  =  (+  5)(+  5)  =  +  25. 

(+3)3  =  (+3)(  +  3)(+3)=  +  27. 

(-f  2)*  =  (+  2)(+  2)(+  2)  (+  2)  =  +  16. 

Evidently,  any  power  of  a  positive  number  is  positive. 

Again,  (_  9)2  =  (_  9)(  -  9)  =  + 81. 

(_6)8=(-5)(-5)(-5)  =  -125. 
(_4)4=(-4)(-4)C-4)(-4)  =  +  256. 
(-  2)6  =(-  2)(-  2)(-  2)(-  2)(-  2)  =  -32. 

Evidently,  any  even  power  of  a  negative  number  is  positive,  and 
any  odd  power  of  a  negative  number  is  negative. 

EXERCISES 

Find  the  products  of  the  following : 


1. 

-1,2,-3. 

7. 

-1,-1,-2,-2. 

2. 

-2,-4,-1. 

8. 

2,-7,4,-2,-1. 

3. 

3,-2,-5,2. 

9. 

-1,-1,-1,-1,-1. 

4. 

2,-5,-8,- 

1. 

10. 

-4,2,-1,-1,-1,-2. 

5. 

-6,-2,-1, 

3. 

11. 

-1,-2,-3,4,1,-5. 

6. 

4,-5,-2,3. 

12. 

2,3,  -1,-2,       1,-2. 

Find  the  values  of: 

13.  (+2/.  19.  (-2/.  25.  (-iy(-2y. 

14.  (-2)^  20.  (-3)^  26.  (-2)2(-3)« 

15.  i-Sy,  21.  (+4/.  27.  (-5)2(+2)« 

16.  (4-1)'.  22.  (-1)».  28.  (_1)«(_2)^ 

17.  (-1>'.  23.  (-2)«.  29.  (-1)5  (-2)3  (-3)2. 

18.  (-4)^  24.  (-3Y.  30.  (  +  2)«(-l)V-3)». 


64  ELEMENTARY  ALGEBRA 

If  a  =  2,  6  =  —  3,  c  =  —  1,  find  the  value  of : 

31.  h^.  34.    2ah\  37.    hh. 

32.  a^h\  35.   3  ac^.  38.   a^  +  h\ 

33.  a6c.  36.    a^bh\  39.    a^  +  ft^  4.  c^  _  a6c. 

35.  Division  of  Positive  and  Negative  Numbers.  —  Division  is  the 
inverse  of  multiplication.  For  example,  to  divide  24  by  8  is  to 
find  a  number  (3)  by  which  8  must  be  multiplied  to  give  24. 
Hence,  the  rules  for  division  of  positive  and  negative  numbers 
come  directly  from  the  rules  for  multiplication. 


Since  (+  5)  (  +  3)  =  +  15,  then  (  +  15) 
Since  (+,  5)(-  3)  =  -  15,  then  (-  16) 
Since  (_  5)(+ 3)  =- 15,  then  (-15) 
Since  (-  5)(-  3)  =  +  15,  then  (+  15) 


(+3)  =  +  5. 
(-3)  =  +  6. 
■(+3)  =  -5. 
(-3)  =  -5. 


Evidently,  from  these  examples  we  get  the  same  laws  of  signs 
in  division  as  in  multiplication  : 

(1)  If  the  dividend  and  divisor  have  like  signs,  the  quotient  is 
positive. 

(2)  If  the  dividend  and  divisor  have  unlike  signs,  the  quotient  is 
negative,  ' 

EXERCISES 


Div 
1. 

ide: 
- 12  by  3. 

11. 

4. 14  by  -14. 

21. 

+  iby-i. 

2. 

-  20  by  -  4. 

12. 

-  20  by  -  20. 

22. 

-4by-|. 

3. 

4- 18  by  -  3. 

13. 

+  50  by  -5. 

23. 

-|by+4. 

4. 

-27  by  +9. 

14. 

-100  by +25. 

24. 

+  iby-f 

5. 

+  64  by  -16. 

15. 

+  66  by  -11. 

25. 

4-fby+}. 

6. 

-81  by  -9. 

16. 

-96  by  -8. 

26. 

-fby-i. 

7. 

+  42  by  -7. 

17. 

-82  by  +41. 

27. 

+  6.4 by  -.8. 

8. 

-54  by  -6. 

18. 

+  63  by  -7. 

28. 

-  2.1  by  -  .3 

9. 

+  72  by  +  8. 

19. 

_84by  -12. 

29. 

-  .63  by  +  .7 

10. 

-  60  by  -  6. 

20. 

+  132  by  +11. 

30. 

-1.44  by -1.2 

POSITIVE  AND  NEGATIVE  NUMBERS  55 

If  a  =  —  2,  6  =  3,  and  c  =  —  4,  find  the  value  of : 

31.  ^.  33.    l^.  35.    2j.  37.    ^. 

c  c  cr  6 

32.  5!^.  34.    ^.  36.    4.  38.    i+*±^. 

36.  Positive  and  Negative  Numbers  in  Equations.  —  In  Chapter 
II  we  learned  how  to  find  the  roots  of  equations  by  means  of 
axioms.  We  may  now  consider  the  members  of  equations  as 
made  up  of  positive  and  negative  terms. 

Let  us  solve  8x  —  7=3x  +  3. 

To  remove  —  7  from  the  first  member  add  +  7  to  both  members. 

Then  8x  =  3a;  +  3+7. 

To  remove  3  x  from  the  second  member  add  —  3  x  to  both  members. 

Then  8x  — 3x  =  3  +  7. 

Uniting  terms,  5  x  =  10. 

Dividing  by  5,  x  =  2,  the  root. 

Observe  that  to  remove  the  known  term  from  the  first  member 
and  the  unknown  term  from  the  second  member,  we  proceed  as 
follows : 

(1)  TJie  known  term  in  the  first  member y  with  its  sign  changed^  was 
added  to  both  members. 

(2)  The  unknown  term  in  the  second  member,  with  its  sign 
changed,  was  added  to  both  7nembers. 

Note.  — The  effect  of  adding  +  7  to  both  members  in  the  equation  above 
was  to  make  the  term  —  7  disappear  from  one  member  and  reappear  in  the 
other,  with  its  sign  changed.  Adding  —  3  x  to  both  members  had  a  similar 
effect.  Evidently,  adding  such  terms  to  both  members  of  an  equation  is 
equivalent  to  moving  terms  from  one  member  to  the  other  and  changing  the 
signs  of  the  terms  moved. 

This  mechanical  process  of  moving  terms  from  one  member  to  the  other  by 
changing  their  signs  is  called  transposition.  The  terms  are  said  to  be  trans- 
posed. Thus,  by  transposing  +  2  and  — 2  a,  a  +  2  =  6  —  2  a,  becomes 
a  +  2a  =  5-2. 

The  process  of  transposition  was  discovered  by  the  Arabs.  About  +  830 
an  Arab  named  Mohammed  ben  Musa  Al  Khowarazmi  wrote  a  mathematical 
book  with  the  title  Aljabr  Walmuqabalah.    The  word  Aljabr  meant  restora* 


66  ELEMENTARY  ALGEBRA 

tion  or  transposition,  as  the  term  is  now  called.     From  this  word  came  the 
name  algebra. 

The  pupil  should  use  the  correct  phraseology  of  "adding  equals  to  both 
members  of  the  equation  "  until  the  thing  actually  done  is  firmly  fixed  in 
mind. 

37.  Equations  with  Negative  Roots.  —  The  roots  of  some  equa 
tions  are  negative  numbers. 

Example  1.  —  Solve  7^  +  15  =  4^  +  3. 
Adding  —  15  and  —  4  ^  (transposing), 
Uniting  like  terms, 
Dividing  by  3, 

Example  2.  —  Solve  3F  +  3  =  8  +  4  F. 
Adding  —  3  and  —  4  F  (transposing),      3  F 
Uniting  like  terms, 
Dividing  by  —  1, 


It- 

-4t  = 

3- 

-15. 

St  = 

— 

12. 

t  = 

— 

4. 

V- 

-4  F  = 

8 

-3. 

-  V  = 

5. 

F  = 

:  _ 

■5. 

EXERCISES 


Solve: 

1.  9y=72/-28. 

2.  3^-4  =  2^-7. 

3.  5  s  +-  20  =  4  -  3  s. 

4.  3m  +  27  =  6-4m. 

5.  5/r-15  =  /r-35. 

6.  2^  +  3  =  4^  +  9. 

7.  7P-8  =  llP+12. 

8.  13  =  3ic  +  40. 

9.  6  -  a  =  12  +-  a. 

10.  17  =  19 +  Jf. 

11.  5  =  5  +  95. 

12.  3-^  =  3^  +  8. 

13.  Tr+7=5  Tf +-20. 


14. 

«-S  =  3. 
5       2 

15. 

8-^  =  10. 
4 

16. 

.  =  1-9. 

17. 

f-f  =  f-' 

18. 

.+|h-io  =  |. 

19. 

^^^^H- 

20. 

^-ttf 

4:  p. 


38.  Interpretation  of  Negative  Roots.  —  In  solving  a  problem  by 
means  of  an  equation,  a  negative  answer  may  have  a  natural  inter- 
pretation,  or  it  may  indicate  that  the  problem  is  impossible. 


POSITIVE  AND  NEGATIVE  NUMBERS  57 

EXERCISES 
Solve  the  following  problems  and  interpret  the  answers : 

1.  On  the  second  of  three  days  the  thermometer  rose  20°  more 
than  on  the  first,  and  on  the  third  day  13°  more  than  on  the 
second.  The  total  rise  during  the  three  days  was  35°.  What 
was  the  change  of  temperature  on  the  first  day  ? 

2.  In  a  certain  year  a  merchant,  on  the  average,  lost  $200  in 
each  of  the  first  5  months,  and  made  a  profit  of  $450  in  each  of 
the  next  6  months.  His  net  profit  for  the  year  was  $  1550.  Did 
he  have  a  loss  or  profit  in  the  last  month,  and  how  much  ? 

3.  A  man  gained  $1300  during  three  months.  During  the 
second  month  he  gained  $650  more  than  the  first  month,  and 
during  the  third  month  $  350  more  than  the  second.  Did  he  gain 
or  lose  during  the  first  month,  and  how  much  ? 

4.  A  farmer  sowed  18  acres  of  wheat  more  the  second  day  than 
the  firsts  12  more  the  third  than  the  second,  and  5  more  the  fourth 
than  the  second.  He  sowed  59  acres  in  all.  How  many  acres 
did  he  sow  each  day  ? 

5.  In  a  triangle  the  sum  of  the  angles  is  180°.  The  second  is 
12°  less  than  one  twelfth  of  the  first,  and  the  third  15°  more  than 
the  first.     How  many  degrees  in  each  angle  ? 

SUPPLEMENTARY   EXERCISES 

1.  Herodotus,  the  Greek  historian,  commonly  called  the 
"  Father  of  History,"  was  born  in  —  484  and  died  in  —  424.  At 
what  age  did  he  die  ? 

2.  Livy,  one  of  the  most  famous  of  the  Eoman  historians,  was 
born  in  —  59  and  lived  to  be  76  years  old.    In  what  year  did  he  die  ? 

3.  A  ship  w^hich  leaves  port  at  42°  north  latitude  goes  one 
day  4°  south,  the  next  3°  north,  the  next  2°  south,  the  next 
3°  south,  and  the  next  1°  south.  If  sailing  south  is  called  —  and 
sailing  north  -|-,  express  the  sum  of  these  motions,  and  find  the 
latitude  of  the  ship  at  the  end  of  the  fifth  day. 


68  ELEMENTARY  ALGEBRA 

4.  A  switch  engine,  making  up  a  freight  train,  moves  forward 
30  yards,  back  120  yards,  forward  150  yards,  forward  40  yards, 
then  back  300  yards.  Kepresent  these  distances  by  4-  and  — 
numbers,  and  find  the  position  of  the  engine  at  the  end  of  the 
work. 

5.  Eind  the  average  of  the  following  temperatures:  —10°, 
-  6°,  2°,  -  4°,  3°,  r,  -  4°,  5°,  16°,  and  - 14°. 

6.  Add  -I,  -J,  +  J,  -i,  and  -\-\. 

7.  Add  + 14|,  -  8|,  -  H,  and  -  5^^. 

8.  rind  the  values  of ; 

(-ly;  (-2)«;  (-2/(-3)3;  (- 5)(-4)X- 3)1 

9.  If  ^  =  -2  and  5  =  -3,  find  the  value  of  (A-B)\ 

10.  If  0?  =  4  and  2/  =  —  2,  find  the  value  of 

(ic^  +  a^  +  ?/2)  (ic2  —  ici/ +  2/2). 

11.  Solve  ^-|  =  |-|.  12.   Solve  5-^=82/4-95. 

13.  Positive  and  negative  numbers  may  be  added  graphically 
as  follows : 

Let  positive  and  negative  numbers  be  represented  by  distances 
measured  to  the  right  and  left,  respectively,  of  a  point  -4  on  a 
line,  as  in  §  28. 


-9-8~?-6-5-4.-3-2-1    0   !   2  3  4  5   6  ^  8  9- 

»  I.-    I         li       A        I         I         I         I    ,1   I         I.    ■  l,riJ  I         I       ■»      ■  I         >         I k. 


To  add  a  positive  number  count  to  the  right  as  many  spaces 
as  there  are  units  in  the  number,  and  to  add  a  negative  number 
count  to  the  left  as  many  spaces  as  there  are  units  in  the 
number. 

Thus,  to  add  —  3  and  +  7,  begin  at  —  3  and  count  to  the  right 
7  spaces.  What  is  the  result?  To  add  —2  and  —6,  begin  at 
—  2  and  count  to  the  left  6  spaces.     What  does  it  give  ? 


POSITIVE  AND  NEGATIVE  NUMBERS  59 

Add  graphically  the  following : 

14.  —5  and  +8.  20.  —2,  —3,  and  +4. 

15.  +  6  and  —  4.  21.  +5,  +2,  and  —  3. 

16.  +2  and  +7.  22.  -6,  +8,  and  -4. 

17.  —  4  and  —  3.  23.  —  1,  +5,  —  7,  and  +  4. 

18.  +4  and  -9.  24.  +  4,  +  3,  -  12,  and  +  1. 

19.  -  7  and  +  12.  25.  +8,  -  8,  -  4,  and  + 10. 

26.  Show  why  a  positive  number  is  subtracted  graphically  by 
counting  to  the  left  the  number  of  units  in  the  subtrahend,  and  a 
negative  number  is  subtracted  by  counting  to  the  right  the  num- 
ber of  units  in  the  subtrahend. 

27.  Subtract  graphically  the  following:  —3  from  —7;  -f  2 
from  —  4 ;  +6  from  —  8 ;  —  5  from  -f-  5 ;  -f  1  from  —  1 ;  —  1 
from  +1;   - 6  from  -6;   +8  from  -2. 

28.  Viewing  multiplication  as  a.  special  form  of  repeated  ad- 
dition, where  the  addends  are  equal,  show  graphically  that  the 
product  of  two  positive  numbers  is  positive,  and  that  the  product 
of  a  negative  number  by  a  positive  number  is  negative. 


CHAPTER   IV 

ADDITION    AND    SUBTRACTION    OF    LITERAL 
EXPRESSIONS 

39.  Addition  of  Monomials.  —  In  §  9  it  was  shown  that  similar 
terms  are  added  or  subtracted  by  adding  or  subtracting  the 
coefficients,  and  attaching  to  the  results  the  common  letters  with 
their  exponents. 

Thus,  3  lb.  +  4  lb.  =  ?  3  ?  +  4  Z  =  ?  6  sq.  yd.  +  10  sq.  yd.  =  ? 

6  2/2  4-  10  !/2  =  ?       9  men  -  4  men  =  ?      9  m  —  4  wi  =  ?      26x^y-  Ux^y  =? 

Since  positive  and  negative  terms  are  by  nature  added  and  sub- 
tracted terms,  respectively,  they  may  be  added  by  the  same  rule  : 
that  is, 

To  add  two  or  more  positive  or  negative  terms,  add  their  coefficients 
according  to  the  rules  in  §  30,  and  attach  to  the  result  the  common 
letters  with  their  exponents. 

For  example,  the  sum  of  —  4  a-6,   +  7  a^ft,  -f  3  a^ft,  and  —  4  a^ft 

-  9  a26  is  _  4  a%  +  7  a^fe  +  3  a2?,  _  9  ^25  ^  _  3  ^25.      The  +  7  a^h 

terms  may  be  written  in  a  column  and   added,  as   in  the  +  3  a^b 

margin.  —  9  a% 

-  3  a^h 
EXERCISES 
Add: 

1.  5a,  4a,  -2a.  7.  -2>x\  -x\V^x\  -5x\ 

2.  9n,  -12n,  -Sn.  8.  PQ,-4:FQ,16PQ, -12  PQ. 

3.  10  ab,  4  ab,  -  ab.  9.   -  F^  9  V,  7  V,  -S  y\ 

4.  6  P,  -  8  P,  -  10  P.  10.  feV,  -  4  6V,  -&V,  -6  6V. 

5.-7  mhi,  4  m^n,  -  5  m^n.  11.  hB\-^  B\  -12  B%  17  B\ 

6.    12^,  -4^, -7^, -9  J..       12.   -^xif,-lx'if,x'if,-2xy\ 

60 


ADDITION  AND   SUBTRACTION  61 


13. 

-4a 

3a? 

-10^ 

15  f 

-20«w 

Qa 

-7aj 

-   8/f        - 

6f 

13  zw 

-8a 

-9a; 

12  K 

f 

ZIO 

14. 

l^xyz 

-    a* 

-12  pq 

100  M 

35  F^ 

- 10  xyz 

7  a* 

-10  pq 

-  30M 

-10  F^ 

—   Qxyz 

-4  a* 

Upq 

-     4.M 

-25  F 

—       xyz 

-     a' 

-      PQ 

M 

-       F 

15. 

orhc 

-      ^H^ 

2ab^ 

-   IT 

-10iV^« 

—   4  a-hc 

-    4s^2 

-  3ab^ 

9r 

4^« 

10  a^hc 

20  s¥ 

(50  ab' 

-16  r 

16iV^« 

-15a'bc 

-   8s«^ 

-  32  ab' 

-       T 

-   3^« 

-      a'bc 

^¥ 

- 15  ab' 

12  T 

-14^« 

16. 

-2MA, 

3.25  A 

20.    ^ 

yy  -\yy- 

fy. 

17. 

0.36  a;,  -3.42  a;. 

21.    1 

«,  -  i  «,  - 

■ia. 

18. 

-1.2  7?,   - 

-  3.4  n,  7.1  n. 

22.    3 

muj  —  ^mrt 

I,  2^  mw. 

19. 

-0.75P«, 

-  2.15  P8,P«.          23.    - 

-2iv",  -i;«,  3|v«. 

Simplify : 

24.  -3a;  +  7a;  +  a;-6a;.  26.    —  5  at^  -  at^ -\- 10  atK 

25.  12c*-{-c*-Sc*-c\  27.   6v'-8v'-10v'4-v'. 

Simplify  by  adding  similar  terms: 

28.  4  a  —  2  a  -  12  a;  +  6  a  +  3  a;. 

Suggestion.  —  Adding  the  terms  in  a  and  the  terms  in  x  separately  gives 
8a- Ox. 

29.  x^  -  ab  -^10  ab  -6  x^  -2  ab, 

30.  3  m-  —  2  mn  —  m^  —  mn, 

31.  a'  +  2ab  +  b'-a'  +  2ab-W. 

32.  y^-3yh-it3yz''-:^-2f-^z^y-4.z^. 


i 


ei  ELEMENTARY  ALGEBRA 

Add: 

33.  4  a  and  —  7  6. 

Solution.  —  Since  these  terms  are  not  simUar,  their  coefficients  cannot  be 
added.    The  addition  may  be  merely  indicated,  giving  4  a  —  7  6. 
Indicating  the  addition  of  dissimilar  terms  is  called  adding  them. 

34.  —  2  a,  3  6,  and  —  c. 

35.  h%  —2,Kk,  and  A:*. 

36.  a^6,  —  ah^j  a^,  and  —  h\ 

Note. — The  process  of  adding  monomials  by  adding  the  coefficients  may 
be  used  to  shorten  computations  in  arithmetical  work.  For  example, 
27  X  54  +  73  X  54  =  5400,  which  is  determined  without  the  aid  of  a  pencil. 

At  sight  give  the  results  of : 

37.  6x48+4x48.  43.     53x84-13x84. 

38.  8x27  +  12x27.  44.    128x96-28x96. 

39.  17x25  +  13x25.  45.  42x28  +  16x28  +  2x28. 

40.  84x63  +  16x63.  46.  34x17  +  27x17-11x17. 

41.  46x35-6x35.  47.  16x38-4x38  +  18x38. 

42.  32x46  +  18x46.  48.  16  tt  +  48  tt  +  36  tt. 

40.  Addition  of  Polynomials.  —  It  is  evident  that  two  or  more 
polynomials  are  added  if  all  of  their  terms  are  added.  Hence, 
two  or  more  polynomials  may  he  added  by  adding  their  similar  terms. 

The  process  of  adding  polynomials  is  similar  to  that  of  adding 
denominate  numbers  involving  two  or  more  units  of  measure. 

For  example  : 

12  bu.  1  pk.  2  qt.  12  6  +    p  +  2  g 

Just  as           5  bu.  2  pk.  5  qt.  so      6&  +  2jo  +  5g 

17  bu.  3pk.  7  qt.  17  6  +  3j9  +  7  ^ 

To  add  polynomials,  ari'ange  the  similar  terms  in  columns  and 
add  each  column  as  in  §  39.  The  sums  of  the  columns,  connected 
by  their  signs,  constitute  the  sum  of  the  polynomials. 


ADDITION  AND   SUBTRACTION  63 

As  pointed  out  in  §  30  (See  Law  of  Order  in  Addition,  §  30, 
Note),  numbers  to  be  added  may  be  arranged  in  any  order  with- 
out changing  the  value  of  their  sum.  In  polynomials  to  be 
added,  if  the  similar  terms  do  not  come  in  the  same  order  in  all 
the  polynomials,  they  should  be  rearrarged  so  that  they  do  come 
in  the  same  order. 

Thus,  to  add  2a  +  36-4c,  5c-4a-&,  and  2a  +  36-4c 

6  4-  5  a  —  2  c,  the  terms  in  the  last  two  expressions  --4a—     b  +  be 

are  rearranged,  and  the  terms  in  a  air^  put  first,  the  5  a  4-     b  —  2c 

terms  in  b  second,  and  the  terms  in  c  hird,  as  in  the  3  a  +  3  6  —     c 
margin. 

EXERCISES 

Add: 

1.   3a-2b  6.       4r-5/-2r" 

_2rH-3r'-7r" 
_9r-    r'  +  4r" 


2. 

4a  +  56 

-ex+    y        * 
4tx-ly 

3. 

4:U  —  2v-\-Zw 
—  2?t-f    V  —  6w 

7.       SA-    B  +  2C 

5A-2B-{-    C 

_    ^4-3^-4(7 


8.  2P  +  4Q4-    E-78 

4.    -3^+    K-12L  -SP-     Q-\-SR-{-2S 

SH-AK-^   SL  7P-6Q-9R  +  SS 
^23-5K-      L 


9.  _20a+3a6-106 

B.    — 8m-f-6n-    p  16a-    ab-\-   4:b 

5m  — 2n4-3p  —   7a  — 5a6+66 

-6m-8n-8p  -   2a-8a&-12& 

10.  2x  —  yj—Sx-^y,  and  Ay  —  Tx. 

11.  5a  +  3c  — 6,  c  — a  +  6,  and  2  6  — 3a-f  4  c. 

12.  S3x^-5xy-^2y^,  Sxy- /  + 6a^,  and  4/- ar'- ay. 

13.  l-2m  +  3m2,  5m-6-i-4m2,  andm2-10m-f-5. 


64  ELEMENTARY  ALGEBRA 

14.  Show  that  the  process  of  adding  polynomials  by  adding  the 
similar  terms  is  used  in  adding  numbers  in  arithmetic. 

Thus,  to  add  723,  422,  and  614,  show  723  =  7  x  100  +  2  x  10  +  3 
that  each  number  may  be  written  as  a  poly-  422  =  4  x  100  +  2  x  10  +  2 
noraial,  as  in  the  margin.  In  what  does  614  =  6  x  100  +  1  x  10  +  4 
this  differ   from  the  ordinary  process  of  17  x  100  +  5  x  10  +  9 

arithmetic  ?  or  1700     +     50+9 

or    1759 

15.  By  writing  them  as  polynomials,  find  by  the  method  in 
Problem  14  the  sums  of  the  following : 

831  123  302  915 

614  502  824  201 

743  954  553  672 

41.  Checking  Work.  —  In  much  of  the  work  of  algebra  one  can 
easily  check  the  results  obtained;  that  is,  determine  whether  or 
not  mistakes  have  been  made  in  the  work.  ^ 

A  good  method  of  checking  addition  of  polynomials  consists 
of  assigning  particular  values  to  all  of  the  general  numbers  in- 
volved in  the  polynomials  and  in  the  sum,  and  seeing  if  the  sum 
of  the  values  thus  obtained  for  the  polynomials  is  equal  to  the 
value  of  the  sum  of  the  polynomials. 

Example.  — Add  and  check :  2  a— 5  6  +  3  c,  3  a+4  6 — 2  c,  and  5  a— 2  b—4c. 
2a  —  5&  +  3c=—  5" 

3a  +  4&-2c=     9         -  -,   r.      »  -, 

^         „i       .  -V   when  a  =  l,  &  =  2,  c=  1. 

5a  — 26  — 4c=— 3 


10  a  -  36- 3c  =  1 
When  a  =  1,  6  =  2,  c  =  1,  the  values  of  the  polynomials  are  —  5,  9,  and 
—  3,  respectively,  and  the  sum  of  these  values  is  1.  The  value  of  the  sum  of 
the  polynomials  also  is  1,  which  shows  that  the  work  is  accurate.  Any  other 
values  of  a,  6,  and  c  might  have  been  used.  Check  this  work  by  giving  a,  6, 
and  c  other  values. 

Note.  —  This  method  may  be  used  in  checking  much  of  the  work  in  algebra 
that  follows.  While  other  methods  of  checking  work  may  be  practiced,  it 
will  be  found  that  this  method  has  the  special  advantage  of  helping  the  pupil 
to  master  more  thoroughly  the  meanings  of  the  symbols  of  algebra.  The 
pupil  should  form  the  habit  of  carefully  checking  all  of  his  work  in  some  way. 


ADDITION  AND  SUBTRACTION  Q^ 

EXERCISES 


Add  and  check : 

1.  2aj2-3a;  +  l 

5a?4.7aj-4 
—  3a^-6a;  +  5 

2.  a3-3a2  +  3a 

4a2-    a 
-a''  +  2a2  +  2a 

-1 
+  3 

6. 

6. 

7. 
8. 

5w  — 6^+   7w7 
—  3tt-2v+   9m? 
-8w-5v  +  11m; 

5r-6s+8 
_7r_9.s-5 
4r  +  8s 

3.  a2  +  2a6+    h^ 
a^^2ab+    b^ 

2a'             -2b' 

4.  26m  +  10n  +  14|) 
—  12m  +  15n-20i) 

-12n-   5/) 

—  5  ci?i  +  3  6w  +  4 

—  3  an  —  5  hm  —  1 

an  +    bm  + 1 
7  a7i  -  2  6w  —  2 

9.  4s  +  3«-4,  3s-2«  +  3,  -s  +  <+5. 

10.  4n2-3ri  +  7,  -  2^2- 9 ?i-3,  6  n^ +  4  w-9. 

11.  6a2  +  5a6-252,  a2-a6-86^  3a2  +  7a6  +  62. 

12.  6a;»-4ar'-3a;  +  8,  6a^  +  3a;+2,  8a^-7a;+9,  a:2_2a;. 

13.  m^— 71^,  2m*— 5mri"+6w',  3?n^n— 7n^,  —  m^— m^n+mn^— n^ 

14.  .4«-2^^  +  5^2_3^  A'-5A'-6A',  9^^  +  7,  -  6  A' 
-  ^2  _  9 

15.  v^-vt-t',  Sir^  +  4:vt-t',  t'-Svt  +  v^,  7vt-St'. 

16.  iax-Tby-cz,  ax-2by,  — 3aa;  +  3c2:,  7by  —  4:CZ. 

17.  pq  +  2qr  —pr,  —  2pq  —  5 qi-  +  4;)r,  7)7  —  pr. 

18.  A:-2F+5F-3,  -3  F+2Ar^+6;fc-3,  2A:2-5,  12+3A:2-^, 

19.  -12r  +  15s-8«,   9r-7s-7^,    — 9r  +  4^,  _54-5r, 
Qr-7t 


66  ELEMENTARY  ALGEBRA 

20.  3v'-5v"  +  v'",  3-4?;',  _  9  v"  +  v'",  7 'y' -  t;^  _6'(;"-hv"'. 

21.  24P+Tr+3^,  5Tr-9^,  6F  +  7H,  -  JI  -  5  P, 
2H-h8W-P. 

22.  5.6a-2.76,  3.5a  +  4.2&,  -2.2a  +  &,  a-6.56. 

23.  .25a;2_i  33.^81^   _ 4.5a;2_i  9^.4^  6.3 -5.8 a;  4-3.5 ic^^ 

24.  5  i22  _  2.5  m,  -  .5  i?^  +  4  m,  1.45  i?^  +  3.2  m,  -  .75  R^-9m. 

25.  1  -  2  2>  -  2.6  i/  +  7.5  Z>^,  !>'  +  5.5  i)^  _  7.2,  -  1.5  + 
6.5  i>  4-  8  B". 

26.  ia;-i2/,  i^j  +  i^/,  aj-y.^ 

27.  \m  —  ^n,  2^m  +  \nj  —\m-\-\n, 

28.  ^H+^K,  1\H  +  ^K,  2H-^K, 

42.  Subtraction  of  Monomials.  —  What  rule  for  subtracting  posi- 
tive and  negative  numbers  was  discovered  in  §  31  ?  Since  a  mo- 
nomial is  itself  a  number,  this  rule  holds  equally  in  the  subtraction 
of  monomials. 

To  subtract  one  monomial  from  another,  change  the  sign  of  the 
subtrahend  and  add  the  result  to  the  minuend. 

The  change  of  sign  of  the  subtrahend  should  be  made  mentally, 
the  written  sign  being  unchanged. 

Thus,  to  subtract  9  x^y  from  —  18  x^y,  let  the  written  work  be  „    / 

as  shown  in  the  margin.  — ^7    2 

—  Zi  X  y 

EXERCISES 


Subtract : 

1.       2a; 

-3a 

-6jr 

8.V» 

-5ab 

-3a; 

-7a 

4^r 

-2f 

-2ab 

2.        4  m^n 

-21xyz 

16^ 

-    6P 

-Sa¥ 

—     m.^n 

—    ^xyz 

-7wH 

-14^ 
-    hk 

-14P 

-    ab' 

3.        2^ 

2abc 

-3^ 

2wH 

-4:hk 

-22  Q 

12  abc 

ADDITION  AND   SUBTRACTION 


67 


4. 

^wv*              5by^ 

-6wv''          -8  6/ 

5. 

From  2  a  take  — 11  a. 

6. 

From  7  a;  take  10  x. 

7. 

From  -6^  take  -4«. 

1. 

4,x-(-Sx)=? 

2. 

-Sy-{-5y)=? 

21x'yh  11 R^        -4V 

-    SxYz        -    3i?3        -Sv' 

8.  From  -  20  P  take  -  9  P. 

9.  From  32  a^^  take  - 16  af. 
10.    From  - 16  ttt^  take  -  7  Trr*. 

13.  5m-(-2?7i)=? 

14.  16^-(-4^)=? 


43.  Subtraction  of  Polynomials.  —  It  is  evident  that  one  poly- 
nomial is  subtracted  from  another  wlien  all  of  its  terms  are  sub- 
tracted. Consequently  for  subtraction  of  polynomials  we  have 
the  following  rule ; 

Add  to  the  min'oend  the  subtrahend  with  the  sign  before  each  oj 
its  terms  changed. 

Similar  terms  should  be  written  in  columns,  as  in  addition. 
The  method  of  checking  work  used  in  §  41  may  be  used  also 
for  checking  subtraction. 

The  change  of  signs  should  only  be  made  mentally. 

Example  1.—  From  Sx^-2xy  +  y^  take  6  y"^  +  4 xy  -{■  S  y^. 
Sx^  —  2xy-\-     y^  =     2 1  when 
6x^  +  4xy-Sy^-=     6  i  a;  =  1 
-  2  x^  -  6  xy  +  4  y'^  =-  4  i  y  =  l 

If  there  is  a  term  in  the  minuend  and  no  term  similar  to  it  in 
the  subtrahend,  such  a  term  will  be  written  in  a  column  by  itself, 
and,  in  subtracting,  it  will  be  placed  in  the  answer  unchanged. 
If  such  a  term  occur  in  the  subtrahend  instead,  in  subtracting,  it 
•»;'iU  be  placed  in  the  answer  with  its  sign  changed. 

BlXAMPLE  2.  —From  4  x*  —  3x4-2  x^ +  7  take  x^-Sx^-{-2x'^  — x. 


2  x«  +  4  x* 


3  x3  +  2  x2 


3x  +  7 

X 


10  1 
1 


«:6+4x*  +  3x8-2xa-2x  +  7  = 


11 


when 
x  =  l 


68  ELEMENTARY  ALGEBRA 

EXERCISES 

Subtract  and  check: 

1.  4ic— 5i/  IZ.   x^  +  x^  -\-x^l 

x—ly  a^  — o^-fo:  — 1 

2.  O  +  7  i) 

3.  4r-12A 


4.  —     m  +  15  a 

—  3m—    4a 

5.  86- 13d 

96+    8d 

6.  r  — 2s  +  5i 
r  +  3g-2^ 

7.  4a;+    2/— 102? 

—  X  — Q>y-[- 11  z 

8.  —  12j9-6g'  +  4r 

—  8y9+     q-2r 

9.  ?^  +     V  —     w 
u  —  2v-\-  Sw 

10.  3/-    c^  +  2^ 

5/-2gr+     h 

11.  20p  —     g  +    r 

2p+  5g  —  7r 

12.  a^  +  2ab-\-b^ 
a^-2ab  +  b^ 


14. 

i^^+  4.F'  +  1 
SF'-'2F'-1 

15. 

2ab-^bc-[-5ac 
6ab-^7bc  —  2ac 

16. 

2P-h4Q-7i? 
_5P_     Q  +  3i? 

17. 

3'y-4'^;'4-10^'" 

_2'y-5'y'  +  12v" 

18. 

7  W-5S-h9H 
2W-i-5S-9H 

19. 

a^^a'  +  a'-l 

20. 

2a^          -5 

21. 

o3_    a2^_|_    ab''-b' 

22. 

6-2m+9m2-m3-7W* 
9+    m  —  9m^-\-m^—m^ 

23. 

a'  +  2ab  +  b' 

24. 

_  6  TFF^  -  F* 

ADDITION  AND  SUBTRACTION 
25.    4-^2_,_7^4 


-e-^ 

if- 

6^ 

26. 

n«- 

3n2 
-    n2 

-2n  +  l 

0-5 

27. 

8a; 
9a; 

-2a;' 

-9 

a;" 
a;" 

28. 

Wf  -  WH  -  10 
-  WH  + 15 

29. 

1.5  a  4- 3.5  6 
.7rt-1.36 

30. 

31.  From  6a-76-10c  take  12  a  +  4  6  -  7  c. 

32.  From  12  ar^  -  3a;  +  1  take  -  2  a;^  +  9  .t  -  7. 

33.  From  m^  —  2  mn  +  ?i*  take  4  m*  —  mn  4-  3  n^ 

34.  From  2-7^2^4^take3^-5^2^8. 

35.  From  2  ^  -h  6  ^2  _^  1  take  9  i^  ^  4  ^  +  3. 

36.  From  15  v'  -  6  -  2  v"  take  _  5  v"  +  9  -  v'. 

37.  From  6  a  -  4  a^  +  7  a^  take  10  -  3  a  +  a^  -  8  a\ 

38.  From  6  P- 5  P^  +  2  P^  take  P^  +  2P2  _  7  p^  3 

39.  From  D"" -\-  R^ -{- 2  BE  take  5  P^  _  7  jr>/^. 

40.  From  a;«  -  3  a;^^,  ^  3  ^2/2  ^^-^^  a^_^r^y_f, 

41.  From  1.5  r  -  s  +  2.8 1  take  .9  i  -  3  r  —  s. 

42.  From  20  ^  -  12.5  E^  +  4.2  take  G  -  2.4  ^  +  9  jE;». 

43.  From  5.6ab  —  Sk  take  4.5  A:  —  25. 

44.  From  :^x^  —  :^-x  —  \  take  2  a;2  _  ^ ^  ^  3, 

45.  m2-5m4-7-(-2m2-3m  +  2)=  ? 

46.  W^-\-24.W-10-(16-5W+W^  =  ? 

47.  «=  -  Z>2  -  (2  a6  -  62)  ==  ? 

48.  6  -  (.S  -  4)  =  ? 

44.  Removal  of  Signs  of  Grouping.  —  When  the  sign  —  precedes 
a  sign  of  grouping  which  incloses  an  expression,  it  indicates  that 
the  expression  is  a  subtracted  expression. 


70  ELEMENTARY  ALGEBRA 

f 

Subtraction  is  performed  by  changiDg  the  sign  of  each  term  of 
the  subtrahend  and  adding  the  resulting  expression  to  the  minu- 
end.    Hence, 

A  sign  of  grouping  preceded  by  the  sign  —  7nay  be  removed  from  an 

expression^  if  the  sign  before  each  term  inclosed  is  changed. 

Eor  example,    3E  -  (5  B -2  S)  =3  R -  6  R  +  2  S. 
And  -  (-  7  1/  4-  2  2/2)  =  7  y  -  2y2. 

Since  the  sign  +  expresses  addition,  or,  when  convenient,  may- 
be considered  merely  as  a  sign  of  distinction,  therefore, 

A  sign  of  grouping  preceded  by  the  sign  -f-  ^^ot?/  be  removed  from 
an  expression,  without  changing  the  sign  of  any  of  the  terms  inclosed. 

For  example,  4w+(3w  —  7)  =  4w  +  3w—  7. 

And  +  (8  -  2  ^)  =  8  -  2  ^. 

When  signs  of  grouping  are  inclosed  within  other  signs  of 
grouping,  it  is  wise  for  the  beginner  to  remove  the  innermost  sign 
first,  then  the  innermost  remaining  sign,  etc. 

Thus,  3a-(9a-{26- [6a-&]}) 

=  3  a  —  (9  a  —{2  6  —  6  a  +  6}),  removing  brackets, 
=  3a—  (9  a  —  26  +  6a—  6),  removing  braces, 
=  3a— 9a4-26  —  6a  +  6,  removing  parentheses, 
=  3  6- 12  a. 

EXERCISES 

Kemove  the  signs  of  grouping  and  add  similar  terms : 

1.  2r+(5r-3).  8.   5 -(4  tt  -  l)-2  tt. 

2.  -%T-(l-2Ty  9.    6i  +  (9-X)+8. 

3.  12  4-{-m;4-w^J.  10.   2  M -\-t-{b  M-t). 

4.  C-[30a-2].  11.   a2-(62_2a&). 

5.  27-(i^-32).  12.    l^f-lyJ^\^f-y\-y. 

6.  a— (6  — 2  a  — 5),  13.    6  w  — 4mn-f  (3  m^i— 2  w). 

7.  2s  +  M^.  14.    2  52_(jg2_4^C'> 

15.    (2r-7^)-(r-4^). 
Suggestion.  —  Since  neither  of  the  signs  of  grouping  incloses  the  other, 
the  tvro  may  be  removed  at  the  same  time,  independently  of  each  other. 


ADDITION  AND  SUBTRACTION  71 

16.  (l-a;  +  a^  +  (3a?-8). 

17.  (2i)-5g)  +  (4i?  +  g)+6g'. 

18.  W-X^w-2.^)-(w-l.bW). 

19.  p_j3Q-li4-J4Q-6i25. 

20.  x'  +  \2x-{y?-^x)\.  23.    -(-(-1)). 

21.  2i;-[«-v+(3^-4^;)].       24.    _(_(_(_(_(- a))))). 

22.  i2D-\^-D\  +  l).  25.    2^-(cZ-(2i-(Z)). 

26.  s-^at+mat-{s-at)\-\at. 

27.  m  — (5n  — (m  — (2w  — 6m))). 

28.  a2~(2  ah  -  (-(-  60))-(«'  +(^'-2  ah)). 

29.  Solve6y-(4-2  2/)=12. 

30.  Solve  F+(3  +  5  F)-7  =  0. 

31.  Solve  12.5  k  -  (10  -  2  fc)  =  19  -  (A;  -  2). 

32.  Solve  32- (8-2)  =  ^-(2  2-10). 

33.  Solvew  =  8-(3i^-14-{6-w}). 

34.  The   sum   of  two   numbers  is   20  and  their  difference  6. 
What  are  the  numbers  ? 

Suggestion.  —  Let  n  =  the  larger  number.     Then  20  —  n  =  the  smaller. 

35.  The  sum  of  two  numbers  is  7  and  their   difference  17. 
What  are  the  numbers  ? 

36.  The  sum  of  the  angles  a  and  h  is  180° 
and  their  difference  30°.  How  many  degrees 
in  each  ? 

37.  My  living  expenses  for  two  years 
amounted  to  $2500.  The  expenses  during 
the  second  year  were  $280  more  than  during 
the  first  year.   Find  the  expenses  for  each  year. 

45.    Insertion  of  Signs   of   Grouping.  —  By  reversing   the   pro- 
cesses in  §  44,  it  follows  that  terms  of  a  polynomial  may  he  inclosed 


72  ELEMENTARY  ALGEBRA 

within  a  sign  of  grouping,  ivhen  this  sign  of  grouping  is  preceded  by 
the  sign  +,  without  changing  the  signs  of  the  terms;  and,  when  pre- 
ceded by  the  sign  —,  by  changing  the  signs  of  the  terms. 

Thus,  to  inclose  the  last  three  terms  of  a^  +  2bc—b^  —  c'^  in  parentheses 
preceded  by  the  sign  +,  we  have  a^  _^  (2  6c  —  &2  _  c2).  If  preceded  by  the 
sign  — ,  it  becomes  a^  ■—  (b^  —  2  be  +  c^). 

EXERCISES 

Inclose  the  last  three  terms  within  a  sign  of  grouping  preceded 
by  the  sign  +  : 

1.  A  +  B-C-D.  6.  x'  +  2xy  +y' -z'  +  2zw-wK 

2.  xy  +  yz-zw-{-wx.  7.  16-8P+P2_^12Jf-9Jf2_4. 

3.  5_2n-9n2  +  n3.  8.  k-{-T-\-10k-l-25k\ 

4.  r._4s  +  4s2  4.1.  9.  IS-B'-b'-Bb. 

5.  m^-}-6pq-q^-9p\  10.  1  +  6rr' -  r^  -  9r'^ 

11-20.  Inclose  the  last  three  terms  in  each  of  the  above  ex 
pressions  within  a  sign  of  grouping  preceded  by  the  sign  — . 

21.  In  an —  2t  —  3n-}-bt,  place  the  terms  involving  n  in  paren- 
theses preceded  by  the  sign  +  and  those  involving  t  in  parentheses 
preceded  by  — . 

22.  In  10 P—  2  W-{-  mP—  n  W,  place  the  terms  involving  P  in 
parentheses  preceded  by  +  and  those  involving  IF" in  parentheses 
preceded  by  — . 

SUPPLEMENTARY  EXERCISES 

Simplify :  ♦ 

1.   5V^  — 8V^  +  6V^  +  Vm  — lOVn. 


2.  Va  +  6  —  4 Va  -h  ^  +  6 Va  +  b  —  9 Va  +  &• 

3.  12(x-y)-3{x-y)-l&(x-y)  +  {x-y). 

4.  _  9(s  -  af^  -  6(s  -  af)  +  (s  -  at~)  +  7(s  -  at"^. 

5.  (2  irR  +  If  +  14(2 ttP  4-  ly  -  8(2  ttP  +  If  +  (27rP  +  7)^. 

6.  -20^/8  +  i%'+6-^8+i%2_^18^8+iTF^2_9^^^|1^^2 


ADDITION  AND   SUBTRACTION  73 

Simplify  by  combining  similar  terms : 

7.  16-2VA-^5VA-24.-7VA. 

8.  D  +  Vw  + -Vi  —  Ww  —  5  D -{- 7 Vw  —  eVi. 

9.  5(W-V)  -SW'  +  (W-V)-\-17  W^-4.{W~-V), 

10.  x^-2xy-\-y^-3x^-6xy-Sy^-{-4:Xy. 

11.  ri^  +  4rir2  -^r,'-2  r,'  -\-Sr,r,  -Sr^ 

Simplify  by  writing  with  a  polynomial  coefficient: 

12.  ax-\-bx  +  ex. 

Solution.  —  These  terms  are  similar  in  ic,  and  hence  are  added  by  adding 
the  coefficients  of  x.  Since  the  coefficients  a,  b,  and  c  cannot  be  added  into 
one  term,  but  their  sum  can  only  be  indicated,  we  get  (a  +  6  +  c)x. 

13.  my-{-ny  —  py. 

14.  al^  -hR^-cBr-\-dR\ 

15.  Mf  +  2Nt^-^  Pe  4-  Qt\ 

16.  5Z)2  +  ^i)2-/iZ)2-7Z>2. 

17.  4:a{x  —  y)—3h{x  —  y)-\-^(x~y). 

18.  Add  nA -\- mB -\- pC  and  xA  —  yB-\-  zC. 

19.  From  all-  hV -  cir subtract  gU  +  hV-  JcWi 
Remove  signs  of  grouping  and  combine  like  terms : 

20.  a  +  [2  a  -  (3  a  -  2  6)]  -  (4  ;>  -  3  a). 

21.  [m  -  (2  m  -  n)]  -  [ -  3  ?i  +  (4  m  -  ?i)]. 

22.  (A-5B)-  \3A-l4.B  +  (6A-B)-(4.B-2A)-]\. 

In  the  following  remove  parentheses,  leaving  brackets,  an^l  Attn 
plify  the  results  as  far  as  possible : 

23.  [(x  -2y)  +  (x  +  y)X(x-2y)-  (x  +  y)l 

24.  [r-(s— «)][r  +  (s-0]. 

25.  [5  A +  (3  +  A)^  [5  ^  -  (3  +  A)]. 

26.  [(4  r  -  3)  +  (3  ?•'  -  4)][(4  r  -  3)  -  (3  r'  -  4)]. 

27.  1(7 a -2b)-  (c  -|- 3 d)][(7 a  -  2  6)  +  (c  +  3 d)'j 


CHAPTER   V 

MULTIPLICATION  AND    DIVISION    OF    LITERAL 
EXPRESSIONS 

46.  Order  of  Factors.  —  Is  there  any  difference  between  the 
values  of  3  X  4  and  4  X  3  ?  Of  2  x  3  x  4  and  3  x  4  x  2  or 
4x2x3  or  2x4x3?  Does  it  change  the  value  of  the  product 
to  change  the  order  of  the  factors  ? 

In  general,  has  ab  the  same  value  as  ba  ?  Has  xyz  the  same 
value  as  xzy  or  yzx  ?  Has  2  x  a  X  3  x  6  the  same  value  as 
2x3xax6? 

Numbers  to  be  multiplied  may  be  taken  in  any  order  without 
changing  the  value  of  the  product. 

Note. — This  fundamental  principle  is  known  as  the  Law  of  Order  in 
Multiplication. 

47.  Grouping  of  Factors.  —  In  3  x  (2  x  4)  the  parentheses  indi- 
cate what  ?  Is  there  any  difference  between  the  values  of 
3  X  2  X  4  and  3  X  (2  X  4)  ?  Of  2  x  3  x  4  x  5  and  (2  x  3)  X  (4  x  5)? 
Does  it  change  the  value  of  the  product  to  group  the  factors  so 
that  the  multiplications  are  performed  in  different  ways  ? 

In  general,  has  abc  the  same  value  as  a  (be)  ?  Has  sc^ocFy^y^  the 
same  value  as  (afx^)  {y^y^)  ? 

Numbers  to  be  multiplied  may  be  grouped  in  any  way  without 
changing  the  value  of  the  product. 

Note.  —  This  fundamental  principle  is  known  as  the  Law  of  Grouping  in 
Multiplication. 

48.  Law  of  Exponents  in  Multiplication.  —  How  else  is  AA 
written  ?  How  else  is  nnnn  written  ?  In  a^  what  does  the  ex- 
ponent show  ?     a^=  aaaaa.     a^  =  ?     i?^  =  ? 

74 


MULTIPLICATION  AND  DIVISION  lb 

Multiplying  a?  by  a^,  we  get 

o}  xa^  =  aax aaa  or  aaaaa  =  a*. 
Similarly,  m*  xm^  =  mmmmmm  =  m^. 

v^  XifX'v^  =  vvvvvvvvvv  =  -y^". 
D'xD'  =  D*+'  =  D\ 

In  multiplying  powers  of  the  same  base,  the  exponent  of  the  product 
is  obtained  by  adding  the  exponents  of  the  factors. 

In  symbols,  the  above  principle  is  expressed  by 

49.  Multiplication  of  Monomials.  —  By  use  of  the  principles 
stated  in  the  preceding  sections  the  product  of  any  two  or  more 
monomials  may  be  found. 

Example  1.  —  To  multiply  2  x'^y  by  3  3?y^^  we  have 

(2  xhi)  (3  7?t)  =  (2x3)  (a;2a^)  (yy^) 
=  6  xV- 

Example  2.  —The  product  of  -  5  M'^N^  and  -  4  M^N^  \a 

(-  6if2JVr6)(-4ilf4iV2)  =  (~  5)(-4)(JJf2JJf*)(iV6iV2) 
=  20  M^N-. 

To  multiply  monomials,  find  the  product  of  the  numerical  coeffi- 
cients, using  the  laws  of  signs  ;  then  annex  to  the  result  the  products 
of  the  like  literal  factors,  using  the  law  of  exponents. 

—    Iv^t^ 
In  performing  multiplication,  the  work  is  often  ^   - 

arranged  in  column  form,  as  here  shown.  ZT^fiw^ 

EXERCISES 

Find  the  product  of : 

1.  a^  and  a^.  5.  'i^  and  ?-*.  9.  y  and  i/^. 

2.  ar'anda;^  6.  Fand  F*.  10.  n«  and  ti^. 

3.  b^  and  b\  7.  U  and  L\  11.  TFand  W^. 

4.  M^  and  M\  8.  P  and  f.  12.  T'  and  T". 


16.    z  and  2;^^ 

19.  p^,  p2^  and  p^. 

17.    A\  A\  and  A\ 

20.    ic,  CC-,  and  x^. 

18.    C\  C\  and  C\ 

21.    w,  71^,  w^,  and  n 

29.    c,  8c2^ 

,  3  c^  and  2  cl 

76  ELEMENTARY  ALGEBRA 

13.  (2^  and  (f. 

14.  s^^  and  s^ 

15.  WsindU^ 

22.  _B,  ^2^  5«,  and  B^. 

23.  2  x^  3  a^,  and  2  ar\  30.  -Sd\  -6d,  d,  and  2  d^. 

24.  4??i,  2  m'*,  and  m".  31.  ic^  and  x^-if. 

25.  -  2  ^,  -  3  ^^  and  4  ^9.  32.  -  2  a?n^  and  5  an^. 

26.  3  jK^  -  7  i?,  and  -  2  R\  33.  -  4  ^(7  and  3  A^B. 

27.  -  5  2/,  -  9  2/^  and  -  2  2/''.  34.  7  s,  -  2  sr,  and  6  s^r^. 

28.  5,-6  5",  and  4  ^.  35.  -  9  a^2/V  ^j^^j  _  4  ^^4^ 

36.  3  Z)W,  -  2  I>W,  and  8  D'^W. 

37.  -  6  M^N^  9  Jf  2jvr  and  -  5  iV^^P. 

38.  16  a;*^^^  -  2  a^;2^  and  7  fz\ 

39.  —  2  mV,  8  my,  and  —  5  ii^p^. 

40.  a^  -2a«62^  7  a^c,  and  -^h(?. 

41.  |-  a^2/>  ~~  i  ^^^  ^11^  ^2/^- 

42.  - 1 HD,  i  H^D,  and  3  Z)^. 

43.  f  rs2,  -^2rV,  and  ir^s^ 

44.  lp(f,  fp^q\  and  2pV. 

45.  F,  iF,  -4F2,  andiF^. 

46.  -  f  cW^,  I  c^d^,  and  cV. 

47.  tV  ^^^'^  ^  TF'^'^  and  4  TF^^ 

48.  -MV%  -\MV\  and  -6il^F2. 

49.  3,-5  Qt,  8  e%  and  -  \  Q^t\ 

50.  Multiplication  of  a  Polynomial  by  a  Monomial.  —  The  multi- 
plication of  a  polynomial  by  a  monomial  is  similar  to  the  multi- 
plication of  denominate  numbers  containing  two  or  more  units  of 

measure. 

9  yd.     2  ft.       6  in.  9  2/-f2/+    6i 

Just  as  4  so  4_ 

36  yd.     8  ft.     24  in.  36  ?/  +  8/+  24  i 


MULTIPLICATION  AND   DIVISION  77 

In  general,  since  the  whole  of  a  quantity  is  multiplied  by  a 
number  when  each  of  its  parts  is  multiplied  by  that  number,  we 
have  the  following  rule : 

To  multiply  a  polynomial  by  a  monomial,  multiply  each  term  of 
the  polynomial  by  the  monomial^  and  add  the  2>CLrii(^l  products 
obtained. 

Note. —This  fundamental  principle,  that  a  polynomial  is  multiplied  by 
multiplying  each  of  its  terms  separately  and  adding  the  partial  products  ob- 
tained, is  known  as  the  Law  of  Distribution.    It  is  expressed  in  symbols  by 

n(a  +  b)=  na  +  nb. 

Example.  — The  multiplication  of  a^  —  2ab  +  b'^  a^  —  2  ah  4-  b"^ 

by  3  a^b'^  may  be  indicated  either  in  the  form  g  ^3^3 

3a«63(a2  _  2  a6  +  b'^)  =  3  a^b^  -  6  a'b^  +  3  a^b^  S  a^b^  -  6  a^b*  +  3  a^b^ 
or  as  in  the  margin. 

It  is  always  more  convenient  in  algebra  to  perform  multiplica- 
tions from  left  to  right.  This  may  also  be  done  in  arithmetic,  but 
for  convenience  in  "  carrying "  it  is  usually  more  convenient  in 
arithmetic  to  begin  at  the  right. 

EXERCISES 

Multiply : 

1.  a; -h  2/ by  5.  11.  a;^  — 2x-|-l  by  4. 

2.  a  —  6  by  3.  12.  a^ -{- 2  ab -\- b^  by  ab. 

3.  n-4by7.  13.  3-8  m  +  4  m^  by  5  m. 
4  P4-lby6  14.  7  a- -^12  b'^  hy  -2  a^b. 

5.  2  ^  -  3  by  4.  15.  15  Rt-A  f  by  4  R\ 

6.  6a  +  5&by2.  16  4.^ +  1  fhj  2s?y\ 

7.  8Jf-9iVby3.  17.  3  s^- 5  s^  +  Q  s  by  4s2. 

8.  -5JE;  +  7  7v^by  10.  18.  B" -A.AGhy  ^  AC, 

9.  4  a; -  9  2/  by  ic.  \2.  5-2  v -ir-1  v^hy  —^v\ 
10.  2  a  -  7  6  +  c  by  6.  20.  X^  -  2  L^  -f-  3  by  4  LK 

21.    3  ?i  -  4  +  12/^  by  -  2  n-p\ 


78  ELEMENTARY  ALGEBRA 

22.  TF^-6  TF2+2  TF-3  by  -5  W\ 

23.  2  c^  + 15  -  4  c^  by  -  3  c\ 

•       24.  A'  +  2AB  +  B'hySA':^. 

25.  a6  +  6c  4-  ca  by  a6c. 

26.  2iy-3DW-\-W'hj  -9L^WK 

27.  ^3-^3  by  2AB. 

28.  Sx^-ea^  +  ii'  +  Sx-lhjSa^. 

29.  7a6  +  3&c  — 4cd  by  2ac.  33.  ^%5p  +  3g). 

30.  5(4- a -a^).  34.  ^(^^ -  5  ^5  +  6  5^)^ 

31.  4:X(3x  +  Sy).  35.  -2  7)1^(3  -  9  m- Sm^). 

32.  Tt(f-at).  36.  3  2/^(a^-3a;^  +  T/). 

37.  -  8a262(_  15  a^b  4.  a?)^  -  b*). 

38.  4  m^n-(m^  —  m^n  +  3  m/i^  —  n^). 

39.   i(4s2-6.s  +  2).  42.    -2a^+^a^?/-ia;?/2+j2/3 

40.  -^w(iw'-iwv+iv^.  -i^y' 

41.    ^a'-^^ab  +  ib'                           43,        1.2  m2- 2.5  mn  + 3.4  ii2 
jaft -1.5m^ 

44.   2.15p3_. 05^  +  1.22) -4 
.5p^ 

Simplify  by  removing  signs  of  grouping  and  uniting  similar 
terras : 

46.   2(2a-36)-3a.  46.   5(A-B)-^2(B- A). 

47.  7(^-2st-\-f)+3{2s^  +  st-^t^, 

48.  a(ab-b^-^b(a^-\-ab). 

49.  3P(5-2P)  +  4(3P+P2). 

50.  iS(2S'-iS)-i-i(2S'-^S). 


MULTIPLICATION  AND   DIVISION  79 

51.  4(2  ^-3  5)- 2(4^  +  ^). 

Note.  —  In  this  expression  the  term  —  2(4  A-{-  B)  may  be  looked  upon  in 
two  ways,  either  as  indicating  that  4  J.  -f  5  is  to  be  multiplied  by  2  and  the 
result  subtracted  from  the  part  that  precedes,  or  as  indicating  that  ^  A  +  B 
is  to  be  multiplied  by  —  2  and  the  result  added  algebraically  to  the  part  that 
precedes.  In  either  case  the  terms  within  the  parentheses  (i:  A-\-  B)  will 
have  their  signs  changed  in  the  process. 

52.  6(c  — 2  A;)  — 4(2  c— 3  A:).  54.    m{n^—mn)  —  n{mn-\-m^). 
53.-  6P-4Q-3(7P  +  2Q).       55.    -5(10r+3  0-2(2  «-r). 

56.  -d(x'y-xf)-(^xy^-o?y). 

57.  -3{^  +  2v^w-2vf)-2{-2i^-\-3v^w-w% 

58.  This  rectangle  is  maxie  up  of  three  parts  whose  common  width 
is  a  and  whose  lengths  are  ar,  y,  and  z,  respectively.  Show  that 
the  areas  of  the  parts  are  ax,  ay,  and  az,  respec- 
tively. What  does  a{x  -\-  y  -\-  z)  represent  ?  ^_^_^_:^ 
Hence,  show  that  a  {x -{- y  -^  z)  =  ax -\-  ay  -{-  az, 
which  illustrates  the  Law  of  Distribution. 

59.  By  a  rectangle  show  in  the  same  way 
that  2x{a-\-2'b  -\-  c)=2ax  -\-4.hx-\-2cx. 

60.  Show  that  the  process  of  multiplying  a  polynomial  by  mul- 
tiplying each  of  its  terms  and  adding  the  partial  products  obtained 
is  used  in  arithmetic  in  the  multiplication  of  a  number  of  two  or 
more  figures  by  a  number  of  one  figure. 

Thus,  to  multiply  432  by  8,  we  see  that  432  is  in  432  —  400  4.  30  +  2 
nature  a  polynomial,  and  may  be  written  as  the  sum  3  3         3       3 

of  three  terms.    432  is  multiplied  by  3  by  multiply-     i296  =  1200  +  90  +  6 
ing  each  of  these  terms  by  3  and  adding  the  results. 

Note.  —  The  ancients  employed  this  principle  in  multiplication.  Thus,  by 
the  Greeks  the  method  of  calculating  7  x  326  was  equivalent  (in  modern  nota- 
tion) to  that  shown  here.  Because  of  the  principle  of 

place  value  in  our  notation,  we  perform  our  multi-     7x326  =  7(300  +  20-1-6) 
plications  mechanically,  without  stopping  to  think  =2100-}- 140 -|-42 

of  this  Law  of  Distribution  underlying  the  work.  —  ^^^^ 

61.  By  the  use  of  a  rectangle  as  in  Ex.  58,  illustrate  the  Law 
of  Distribution  in  finding  3  X  268 


fp^ 


^ 


fz 


80  ELEMENTARY  ALGEBRA 

51.  Multiplication  of  a  Polynomial  by  a  Polynomial.  —  The  pro- 
cess of  multiplying  one  polynomial  by  another  is  similar  to  the 
process  iu  arithmetic  of  multiplying  one  number  of  two  or  more 
figures  by  another  number  of  two  or  more  figures.  Just  as  in 
arithmetic  the  multiplicand  is  multiplied  by  the  number  repre- 
sented by  each  figure  of  the  multiplier,  and  all  the  partial  prod- 
ucts added,  so  irf  multiplication  of  polynomials  in  algebra  the 
^uultiplicand  is  multiplied  by  each  term  of  the  multiplier,  and  all 
^he  partial  prod»^.ts  added. 

For  example : 

26  2a;+5y 

34  3a;-f4y 

^ust  as      100  =    4  X  25  so      6 x^  +  \^xy  =  3 a;(2  a;  -}-  5y) 

750  =  -SO  X  25  8 xy  +  20  y~  ^4.y(2x -\- by) 

850  =  S4  x25  6x2  +  23icy-F20?/-^  =  (3a:  +  4?/)(2x  +  5?/) 

Each  term  of  2  x  -|-  5  ?/  is  multiplied  by  3  a;  and  then  by  4  y,  and  the  similar 
^rms  of  the  partial  products  are  written  in  columns  and  added. 

Note.— See  that  pupils  clearly  understand  that  we  use  this  process  abridged 
f.n  arithmetical  multiplication.  Then  have  them  see  that  algebraic  multipli- 
cation, is  the  same. 

It  is  easily  seen  that  this  process  is  only  an  application  of  the 
Law  of  Distribution  noted  in  §  50.     Hence, 

T'o  obtain  the  product  of  two  polynomials,  multiply  the  multipli- 
cand by  each  term  of  the  multiplierj  and  add  the  resulting  partial 
products. 

The  work  in  multiplication  can  be  checked  by  assigning  particu- 
lar values  to  the  general  numbers  involved,  as  in  addition  and 
subtraction,  and  seeing  if  the  product  of  the  particular  values  of 
the  factors  equals  the  particular  value  of  the  product. 

Example  1. — Multiply  2a— 36+5cby3a+6— 2  c. 

^a~+     h  -2c  =^ 

6  a2  -  9  a&  +  15  ac 

2a6  _  3  62+    5  5c 

-    4ac +    6  6c  -  10  c2 

6a2  -  7  a6  +  11  ac- 3  62  +  11  6c  -  10c2  =  6 


MULTIPLICATION  AND  DIVISION  81 

When  a  =  1,  &  =  1,  and  c  =  1,  the  multiplicand  =  4,  multiplier  =  2,  and 
product  =8,  as  it  should. 

Observe  that  the  terms  of  the  first  partial  product  are  written  in  a  line  by 
themselves,  those  of  the  second  in  a  line  by  themselves,  etc.,  and  that  the  new 
kinds  of  terms,  when  obtained,  are  spaced  out  so  that  only  similar  terms  will 
fall  in  columns. 

Note. —  A  polynomial  is  said  to  be  arranged  according  to  the  powers  of  a 
letter  when  the  exponents  of  that  letter  either  increase  or  decrease  in  the  suc- 
cessive terms  as  we  pass  from  left  to  right.  Thus,  o^  —  2  x^  +  x^-f-  x  —  5  is 
arranged  according  to  the  descending  powers  of  x ;  while  2  y*  +  xy^  +  x'^y^ 
-f  x^y  +  x^  is  arranged  according  to  the  ascending  powers  of  x.  How  is  the 
latter  arranged  as  to  y  ? 

It  will  be  found  an  advantage  in  multiplication  of  polynomials  to  arrange, 
if  possible,  both  the  multiplicand  and  the  multiplier  alike  according  to  the 
powers  of  some  letter,  before  multiplying. 

Example  2.  —  Multiply  A^  -  2  ^  2  A\)j  A-  Q  +  A^. 
Arranging  both  trinomials  according  to  the  descending  powers  of  A^  we 
have 


A^^2A   -2                             =      1 

A^-{-     A  -Q                            =-4 

^*  +  2  ^3  -  2  ^2 

when 

A^  +  2A^-    2  A 

^  =  1 

-QA^-\2A  +  \2 

^4  +  3  ^a  _  0  ^-2  _  14  ^  +  12  =  _  4 

Note.  —  Substituting  1  for  each  of  the  letters  in  the  two  preceding  examples 
tests  the  coefficients  and  signs  only.  Why  ?  To  test  the  exponents  other 
values  than  1  must  be  assigned  to  the  letters. 


EXERCISES 

Multiply  and  test : 

1.  ri  +  2byn  +  2.  7.   8&  +  lby36  +  l. 

2.  A-^hy  A  +  2.  8.   3  72  +  1  by  4  i? +  1. 

3.  a;-4bya;-3.       .  9.   2F+5by3F+7. 

4.  2a-f3  by  3a-2.  10.   5^/- 8  by  6?/- 1. 

5.  5  ;z  4- 6  by  2 +  4.  11.   4  Z)  +  2  by  -  3  Z>  +  5. 

6.  4  i  -  7  by  4  ^  4-  3.  12.2  m  +  6  by  2  m  -  6. 


82  ELEMENTARY  ALGEBRA 

13.  a-lObya  +  lO.  21.  2A:  -  3c  by  2  A:- 3  c. 

14.  2  TF  +  3  by  2  TF+  3.  22.  x" -2  x -^Ihy  x -{-  2. 

15.  3  +  4a;by  2  +  3a;.  23.  A^ -^  A  +  4.hy  A -h. 

16.  7  —  /S'  by  3  +  /S'.  24.  '?;2  +  'yw  +  w^  i^y  ^  ^  ^^ 

17.  5  _  4  5  by  5  -  4  6.  25.  Jf^  _  ^2  ^y  jj^2  ^  ^2^ 

18.  2a  +  36  by  5a  — 2&.  26.  2a;— 32/  +  1  by3x+22/— 1 

19.  Q>p  +  4:thj  p  —  2t.  211.  a;2  —  4  +  3  ic  by  2  +  5  a;. 

20.  ^B  +  GhybB+G.  28.  6- 5  72  by  4  i2  + 7. 

29.  a^-a-  +  4a-6by  2a2-3a  +  2. 

30.  3  71^  —  71  +  4  by  4  71  -  3. 

31.  l-2>S  +  3>S'2by4^  +  5. 

32.  24-3r+7''by  2  +  3r+T2. 

33.  4a  +  66  +  10cby  2a-36+4c. 

34.  R-bS+'dWhj3R-\-2W-S. 

35.  ia  +  ibyia  +  i- 

36.  2  71  +  6^- 5r  by  2p  — 3n. 

37.  a  —  1  by  1  +  2  a  +  a^. 

38.  36^-562  +  &_4by  3-2&. 

39.  D-2D^  +  4.hy%-\-2D-D\ 

40.  2w3-5w;2^4w-l  by  3-2^2  — 2(;. 

41.  (a? +  5)  (2  a; -5).  46.  (3  -  v)  (v^  +  6  -  v). 

42.  (2  ^  +  4)  (5^-1).  47.  {a^^Jf){a^-W). 

43.  (1-40(1+40.  48.  (7i2-4)(7i^  +  6  7i2  +  5). 

44.  (S2-4^(7)(jB2+4^0).  49.  {a  -  ^  t)  {2  a' -\- at  -  f), 

45.  (a^-2a;  +  l)(aj-l).  50.  {\x  +  \y){^x-\y). 

Find  the  product  of ; 

51.    a;  —  1,  0/' +  2,  and  a; -}- 1. 

Suggestion.  —  Multiply  x  —  1  by  a;  +  2,  and  that  product  by  x  +  1.     .. 


MULTIPLICATION  AND  DIVISION 


83 


52.  a  +  3,  a  —  2,  2  a  + 1. 

53.  2J5-1,  J5  +  2,  3JB-2. 

54.  3F-4,  3  F-4, 1  +  F. 

55.  m  4-  n,  m  —  w,  m^  +  n^. 


56.  2  4-d,  2-d,  4  +  (2l 

57.  2p  —  ^t,^p-\-2t,t-\rp, 

58.  ^2^^4-1,  ^_  1,^3 _^i^ 

59.  a^  —  2^,  (c^  4-  z^y  ic^  +  2^. 


60.  (^  +  l)(^  +  2X^  +  3)(^  +  4). 

Bemove  signs  of  grouping  and  simplify  by  combining  terms : 

61.  (i4-2)(^  +  3)  +  (2i-l)(«-4). 

62.  (2  4-F)(3i^-l)+i^(5i^-2). 

63.  h''-2h  +  {l-h){\+h). 

64.  {x  4- «;)  (a;  -H  to)  -\-(x  —  w)(x  —  w), 

65.  (a  +  2)(a4-5)-(2a4-l)(a-2). 

Suggestion.  —  The  product  of  (2  a  +  l)(a  — 2)  is  to  be  subtracted  from 
(a  +  2)  (a  +  6) .    What  operation  must  be  performed  first  ? 

66.  (A-h  B)(A  +  E)  -  (A-  B){A-  B). 

67.  2(2/^-32/  +  !)- (2/ -f4)(2/-l). 

68.  (v-^t)(v-2t)-(v-t)(v-{-2t). 

69.  n2-3w  +  4-/2-w^(3?i4-2). 

70.  (2  a  4-  3  6)(2  a4-3&)-(2a-3  6)(2  a  -  3  6). 


o?  +     ^ 


71.  Show   by   the    accompanying   diagram 
that  (a.  4-  b)  (c -\- d)  =  ac -\- be  +  ad -\-  bd. 

72.  Show  by  a  diagram,  as  in  Problem  71, 
that  (a;  4-  5)  (a;  4-  7)  =  ar^  4- 12  a;  +  35. 

73.  Show  by  a  diagram,  as  in  Problem  71, 
what  the  product  of  (a-4-  &  4-  c)  (a;  -h  i/  +  2^)  is. 

74.  Show  how  the  Law  of  Distribution  is  used  in  multiplying 
the  following : 


{ 

c?<r 

3c 

i 

ad 

6d 

234 
42 


506 
63 


427 
135 


84  ELEMENTARY  ALGEBRA 

75.  A  rectangular  field  is  16  rods  longer  than  it  is  wide.  A 
second  field  is  5  rods  shorter  and  2  rods  wider  than  the  first. 
If  w  represents  the  width  of  the  first,  express  the  area  of  the 
second. 

76.  A  rectangle  is  8  inches  longer  than  it  is  wide.  If  its  length 
is  increased  by  4  inches,  and  its  width  decreased  by  2  inches, 
represent  by  use  of  one  letter  its  resulting  area.    . 

52.  Law  of  Exponents  in  Division.  —  Since  division  is  the  inverse 
of  multiplication,  the  law  of  exponents  in  division  is  obtained 
from  the  law  in  multiplication. 

Since  a^  y.  a!^  =  a^,  a^  -^  a^  =  a^. 
Since  w*  x  w^  =  n^^^  n^^  -^  n^  =  n^. 

Tell  how  the  exponent  of  each  quotient  is  obtained  from  the  exponents  of 
the  dividend  and  divisor. 

By  use  of  cancellation  we  arrive  at  the  same  conclusion. 

For  example,  ^'  =  #^  ^  a:a:xa;  =  x\ 

x^         ^jt 

In  dividing  poiuers  of  the  same  base,  the  exponent  of  the  quotient  is 
obtained  by  subtracting  the  exponent  of  the  divisor  from  the  exponent 
of  the  dividend. 

In  symbols  this  rule  is  expressed  by 


n  —  ffm~n 


53.  Division  of  Monomials.  —  By  reversing  the  rule  for  multi- 
plying monomials  the  rule  for  division  is  obtained. 

Since  2  a  x  S  b  =  6  ab,  6  ab  -^  2  a  -  S  b. 

Since  (7a:3)(_3a;5)^_2ia:8,  {  ~  2lx») --- (^7  x^)  =  -  S x^ 

In  dividing  monomials,  divine  the  coefficient  of  the  dividend  by  that 
of  the  divisor,  using  the  laws  cf  signs;  then  divide  the  literal  factors 
by  subtracting  the  exponents  oj  the  letters  in  the  divisor  from  the  ex- 
ponents of  the  like  letters  in  the  dividend. 

Thus,  (6  x^f)  ^  (  -  2  x?/4)  =  -  3  x^y"^. 


MULTIPLICATION  AND  DIVISION  85 

Evidently,  if  a  letter  has  the  same  exponent  in  the  dividend 
and  divisor,  this  letter  will  not  appear  in  the  quotient.     Why  ? 

Thus,  (-24a:3^)-=-(-4a;3y4)  =  6  2/2. 

Here  x  does  not  appear  in  the  quotient. 

This  division  may  be  indicated  also  in  the  form 

-^x^t  -4a8y4)-24xV 


EXERCISES 

Divide : 

1.  a^  by  a\  9.  s»  by  s«.  17.  a?h^  by  a^h\ 

2.  N^  by  N\  10.  P^  by  P^'.  18.  x'lf  by  ar^?/'- 

3.  ^  by  i?2.  11.  ic^8  by  3?.  19.  m^?*^  by  mV. 

4.  y^hjf.  12.  ^l^by^^  20.  a^ft V  by  a262c2, 

5.  ^by«2.  13.  m^^hym'^.  21.  6a^by2a^. 

6.  /ifi"  by  K\  14.  Z)<  by  D\  22.  8  iJv^  by  2  i??;^. 

7.  v'byv^  15.  5^by6^  23.  4  ttTJ^  by  2  ttT?. 

8.  afi  by  ar^.  16.  z"  by  2;.  24.  18  TF/  by  9  TF. 

25.  9a8by-3a''.  33.  - 50 a^d*  by  2 rc^^ 

26.  15  ^^  by  -  5 1^.  34.  ^A^hy  ^  A\ 

27.  -  24  P*n  by  -  6  P.  35.  6  zH  by  4  z^t 

28.  -  56  Jtf  ^iV^3  by  7  jvr2.  36,  __  2''mPn  by  3  m^/i. 

29.  36a^6V  by  -  9a26V.  37.  J  Q<^  by  2  Qt^ 

30.  4  xV  l^y  -  25*/.  38.  J  ««/2«  by  - 1  x^i/*- 

31.  -  6  Vt^  by  2  FiJl  39.  7  7^2^j4  by  -  ^  n\ 

32.  18  Z)*v  by  6  D^  40.  -  a;i>5  by  - 1  a;Z)3. 

54.  Division  of  a  Polynomial  by  a  Monomial.  —  The  division  of  a 
polynomial  by  a  monomial  is  similar  to  the  division  of  denominate 
numbers  containing  two  or  more  units  of  measure. 

5  ft.    3  in.  5/+3i 

•^^^'^  8)15  ft.     9  in.         '^  3)16/+ 9^ 


86  ELEMENTARY  ALGEBRA 

In  general,  since  divisor  x  quotient  =  dividend^  the  quotient  wiust 
be  such  that  the  product  of  its  terms  by  the  divisor  will  give  the 
terms  of  the  dividend.     Hence  the  rule : 

To  divide  a  polynomial  by  a  monomial,  divide  each  term  of  the 
dividend  by  the  dioisor,  and  add  the  partial  quotients. 

Example.  — Divide  4:ofiy  —  S  xV  +  6  x?/^  by  —  2  xy. 

We  divide  each  term  of  the  dividend  hj  —  2xy  and  set  the  results  down 
in  a  line  with  their  signs,  which  indicates  their  sum.  The  work  may  be 
written  in  either  the  form 

(4 x^y  -  8 x2?/2  4-  6 xy'^) -i-(-  2  xy)  =  -  2  x^  +  4 xy  -  Sy^ 
-2x2  +  4xy- 3y2^ 
or  —  2  x?/)4  x^y  —  8  x'-^?/"-^  +  6  xy^ 

EXERCISES 
Divide  : 

1.  4m-67iby2.  6.  -24P+ 72  Q  by  -  12. 

2.  12D-{-9FhjS.  7.  81  a6  -  18  ac  by  9a. 

3.  16a;  +  24by4.  8.  a;  +  2a;2bya;. 

4.  21a-286by7.  9.  A'-SA'hjA\ 

5.  36  +  422/ by  6.  10.  30 m^  +  40 m^w  by  5 m*. 

11.  2pq  —  4:pt-]-6pvhj  2p. 

12.  _  6  TTV  + 18  Wy- 12  Wy  by  6  W^g. 

13.  af-x^bja^.  15.    27^-18^  by  -  9^. 

14.  a*  —  a^b -j- a%^  hy  a^.  16.    — x  —  y-\-zhj—l. 

17.  -A-^2B-Chj  -1. 

18.  4  w; V  +  8  w;3^3  _  12  wh*  by  4  w;^?;^. 

19.  12 pH  -  8i>¥  +  24p2  by  4^1 

20.  -21a^-35a^-l4aj^by  -7a?*. 

21.  (8a6-46c-206c?)--4&. 

22.  (2;^a;?/  —  2  giya^  -f-  3  2?/^)  -=-  21/. 

23.  (2P¥-3P¥-f-P^^)--P2^. 


MULTIPLICATION  AND  DIVISION  87 

24.  {-a^-to?Jrfa)^{-a). 

25.  {M^N^-^MN^)^{-MN^). 

26.  {a?K-lh^K+2cK)^{-K). 

28.    {-^xy-\Jy)^{-\xy). 

29-  (i%-iTr/)^(i%). 

30.    (.5«  +  «'-1.5^)-i-(.5  0. 

55.  Division  of  One  Polynomial  by  Another.  —  Let  us  divide 
a^  +  a;4  +  7a^-6a;  +  8bya;--|-2a;  +  8. 

First,  arrange  both  dividend  and  divisor  according  to  the 
descending  powers  of  x  (see  §  51).  The  work  may  be  arranged 
ag  below.  2,1 

x2  +  2  X  +  8)a;*  +     x«  +  7x2_0a;  +  8 
a^  +  2  a^  +  8  x^ 

-  x*-    x2-6x  +  8 

-  x3  -  2  x2  -  8  X 

x2  +  2  X  +  8 
x2  +  2x  +  8 

Explanation.  —  The  product  of  the  term  of  highest  power  in  x  in  the 
quotient  and  the  term  of  higliest  power  in  the  divisor  must  give  the  term  of 
highest  power  in  the  dividend.  Hence,  the  highest  term  of  the  quotient  is 
obtained  by  dividing  the  highest  term  of  the  dividend  x*  by  the  highest  term 
of  the  divisor  x^.     This  gives  x-,  the  first  term  of  the  quotient. 

Multiply  the  whole  divisor  by  the  term  of  the  quotient  just  found.  This 
gives  X*  +  2  x^  +  8  x2,  which  is  placed  below  the  dividend. 

The  dividend  is  the  product  of  the  divisor  by  the  whole  quotient.  Hence, 
subtracting  the  product  x*  +  2  x^  +  8  0-2  from  the  dividend,  the  remainder 
—  x'  —  x2  —  6x  +  8  must  be  the  product  of  the  divisor  by  the  part  of  the 
quotient  to  be  found. 

Therefore,  the  product  of  the  next  highest  term  of  the  quotient  by  the 
highest  term  of  the  divisor  must  equal  the  highest  term  of  the  remainder. 
Hence,  dividing  —  x^  of  the  remainder  by  x'-^  of  the  divisor  gives  —  x,  the 
second  term  of  the  quotient. 

Multiply  the  whole  divisor  by  the  new  term,  —  x  :  subtract  the  product 
from  the  remainder.     This  leaves  x-  +  2  x  +  8. 


88  ELEMENTARY  ALGEBRA 

Evidently  the  third  term  of  the  quotient  will  be  obtained  from  this  second 
remainder  just  as  the  second  term  was  obtained  from  the  first  remainder. 
By  continuing  this  process,  all  of  the  terms  of  the  quotient  may  be  found. 

If  in  any  problem  the  divisor  is  an  exact  divisor  of  the  dividend, 
the  work  may  be  carried  on  until  a  remainder  zero  is  found. 
Otherwise  the  work  may  be  continued  until  a  remainder  is  ob- 
tained in  which  the  highest  term  is  of  lower  power  than  the 
highest  term  of  the  divisor.     This  is  a  t7me  remainder. 

Hence  the  rule  for  dividing  one  polynomial  by  another : 

1.  Arrange  both  the  dividend  and  divisor  according  to  the  descend- 
ing or  ascending  poivers  of  some  letter. 

2.  Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor  to  obtain  the  first  term  of  the  quotient. 

3.  Multiply  the  whole  divisor  by  this  term  of  the  quotient,  arid 
subtract  the  result  from  the  dividend. 

4.  Treat  the  remainder  as  a  new  dividend  (having  the  terms 
arranged  as  before),  and  repeat  the  process,  continuing  until  either 
the  remainder  zero,  or  a  true  remainder,  is  found. 

Example  1.  —  Divide  a^  —  11  a  +  30  by  a  —  6. 

a-6 

a  -  6)a2  -  11  a  +  30 
a2-   5a 

-  6a +  30 

-  6a +  30 


Check.  —  When  a  =  1,  dividend  =  20,  divisor  =  —  4,  and  quotient  =  —  5, 
as  it  should. 

Example  2.  —  Divide  x^  -{■  ij'^  hy  x  +  y. 

3-3  _  x^y  _j_  3-^2  _  yZ  


X  +  y)x^ 

+ 

y' 

x^  +  x^y 

-x^y 

+ 

y' 

—  x^y- 

-  T^Y 

x^y'         + 

y" 

x2y2  +  xy^ 

-xy^  + 

y' 

^xy^- 

jl 

2  1/4 


MULTIPLICATION  AND  DIVISION  89 

In  this  example  there  is  a  true  remainder  2  y^  obtained.  By  using  the 
fractional  form  to  indicate  the  division  of  the  remainder,  as  in  arithmetic, 
the  entire  quotient  in  this  problem  may  be  expressed  thus: 

x-\-y  x  +  y 

Check.  —  When  x  =  2  and  y  =  1,  dividend  =  17,  divisor  =  3,  quotient 
=  6|,  as  it  should. 

Note. — Observe  that  the  terms  in  the  dividend,  divisor,  and  all  the  prod- 
ucts and  remainders  in  the  above  example  are  arranged  in  descending  powers 
of  X.  Similar  terms  should  be  kept  in  columns  in  the  work.  For  this  pur- 
pose it  is  often  necessary  to  leave  vacant  spaces  between  the  terms,  as  in 
this  example. 


EXERCISES 

Divide  and  check : 

1.  a^  +  3a  +  2hy  a-{-l.  14.   4  ^2_  g  ^^  2^-f  3. 

2.  iV2  +  3i\r_40  by  JV^+8.       15.   25  R'- A  by  5  M -2. 

3.  a^-5x  +  6hyx-2.  16.   4.p^ -Ihy  2  p-{-l. 

4.  p2-f-7P-f-12by  P+3.  17.   l-da'hyl-Sa. 

5.  t^-6t-\-Shy  t-4:,  18.    16  -  25  D^  by  4  +  5  Z). 

6.  y^-9y-\-20hy  y-4:.  19.   a^-b'^hy  a-b. 

7.  H'^H-SOhy  H-S.  20.   x'-4:fhyx-^2y. 

8.  &=»  +  &-20by  6  +  5.  21.    TF^- 25  /  by  TT-S^. 

9.  m^  +  3  m  —  40  by  m  — 10.       22.    7n^-\-5  mw  +4  ?i-  by  m  +  4  n. 

10.  2a^  +  3a:  +  lby  2a;  +  l.         23.    P'-9 PQ-\-20Q- by  P-5  Q. 

11.  3a^  +  16fl;-12by  a;-|-6.        24.   (c^  ^  21 -f  10  a;  by  3  +  «. 

12.  12Jr2-^-6by  4/r-6.       25.    1 +  2  a-f  a^  by  a  +  1. 

13.  6  F2+29F+35by  3  F+7.   26.    15  2/  +  36  +  / by  2/  +  4. 

27.  12^-30  J5:2^2J5;-5by  2^-5. 

28.  2r3-3r'  +  3  +  4rby  3  +  2r. 

29.  ^  - 1  by  5  -  1. 

30.  f-^by  y-\-z.  32.    H'' -fby  H^ -\-f, 

31.  W^-V'by  W^+V^.  ^^.   14AR''-lbyl2R  +  l. 


90  ELEMENTARY  ALGEBRA 

34.  63^3&2^_36  +  iby62  +  2&  +  l. 

35.  ic*  4-  x^  +  1  hj  x^  —  x^  +  1. 

36.  n^  —  a^  by  )i^  —  a^. 

37.  22/^-92/^^-14  2/4-17  2/' by /-2y. 

38.  8d^  +  27«^by  2d-3^. 

39.  4g*-9g2_|_20g-25  by  2^2 _^Sg-5. 

40.  ^3-2.42  +  5  by  A-5. 

41.  ^^-3/A:  +  6A;2by^2_3;^^ 

42.  a'*  +  7i^  +  a^n^  by  n^  -}-  a^  —  an. 

43.  16  7i^-96/i3  +  81-216/H-216/i2  by  94-4/^2-12^. 

44.  A  rectangular  field  whose  area  is  oj^  4-  3  a^2/  +  ^  ^V^  +  2/^  ^^s 
a  length  of  ic^  4-  2  ^2/  4-  2/'.     What  is  its  width  ? 

^N^^  Show  that  the  problem  682  -;-  31  may  be  written  in  the 
f6£hl  (6^2_^8«4-2)-^(3f  4-1).  Eind  the  quotient  in  this  form, 
and  compare  the  answer  with  that  obtained  by  the  arithmetical 
process. 

46.  In  the  problem  882-5-42  express  the  given  numbers  as 
polynomials  and  find  the  quotient.  Compare  the  answer  found 
with  that  obtained  by  the  arithmetical  process. 

47.  Divide  1  -J-  a;  by  1  —  a;.  How  many  terms  are  there  in  the 
quotient? 

48.  Divide  1  4-  2  n  by  1  4-  n,  carrying  the  quotient  to  four  terms. 

49.  Divide  1  4-  2/  by  1  —  y,  carrying  the  quotient  to  four  terms. 


^/J 


SUPPLEMENTARY  EXERCISES 

Find  the  product  of: 

1.  a^  —2a^,  and  —  3  a.  6.  a^ft^+i  and  a^+^ft". 

2.  x'^  and  o^".  7.-8  W^v  and  3  TfV. 

3.  2/^  2/^^  and  2/*"*.  8.  52*^,  -^22"+^  and  2-^ 

4.  3  r,  —  4  r+i,  and  —  2  r-\  9.  a''  -  a""  -  oT-  ■  a""  -  a". 
5.-5  7^«+^  —  2  i^'*-^  and  R^.  10.  aj*"  •  a?*"  •  a;"*  •  a?"*  •  •  •  to  w  factors 


MULTIPLICATION  AND  DIVISION  91 

Divide : 

11.  w^''  by  w\  15.  12a^  by  -Q^x''. 

12.  Jtf «  by  ilf*.  16.   -  63  ^^  by  -  7  A\ 

13.  2/'"'  by  2/2«».  17.  43  a;«+^6'-'*  by  -  6  a;*5»-^ 

14.  JB«+^  by  ^.  18.  —  80  vVT  by  16  v'^-H''-\ 

Multiply : 

19.  a"  +  &''  +  c''  by  a^  22.  P"4-  Q"  by  P  +  Q". 

20.  a;'^^  —  2/"  +  2;""^^  by  m^f.  23.  a^  +  h^  by  a^  —  6". 

21.  ic«  —  I/*  by  af  — 1^/^  24.  a;"—  ?/",  a;"  + 1/",  and  a^  H- 2/**. 

25.    F'4-1,  F*-l,  F^  +  1,  F^  +  1. 

Divide : 

26    a^  —  a^  by  a*. 

27.  10a"+2-4a''+»by  2a2. 

28.  Jf^  +  ^f+^  +  itf^+^by  iJf'. 

29.  2/"+ V*+^  +  2/2*»+ix'*+^  by  a;«+V^*« 

30.  a^  —  2/"  by  a^  —  2/^. 

31.  "nt^  _  ];^4«  by  T^"  —  F**. 

82.  a«"»4-6^by  a"'  +  &"*. 

83.  /S'*«-  T^-by  S"-  T". 

84.  a;3^3a,2y_^3a^^2/3_|.^bya;  +  2/  +  2. 

Suggestion.  —  Arrange  the  dividend,  divisor,  all  partial  products,  and 
remainders  according  to  the  descending  powers  of  x,  without  reference  to 
the  other  letters  involved.  If  two  or  more  terms  in  any  case  contain  the 
same  power  of  x,  it  does  not  matter  which  precedes. 

35.  a^-\-b^-c^-^3abchy  a-{-b-c. 

36.  m^ -\- n^ -\- p^  —  3  mnp  hj  m -\- n -{- p. 

37.  u^  +  v^  +  iv^ -\- 2  uv -{- 2  uw  -\-2vwhj  u-\-v-\-w. 

38.  4 a^  +  4 a6  +  6^ - 12 ac- 6 6c +  9 c^  by  2a. +  6 -3c. 


CHAPTER  VI  1/ 

LINEAR  EQUATIONS:    PROBLEMS 

56.  In  Chapters  II  and  III  we  found  how  to  solve  problems 
by  means  of  equations  containing  one  unknown  number.  With 
a  knowledge  of  the  principles  discovered  in  Chapters  IV  and  V  at 
command,  we  are  now  able  to  consider  additional  points  connected 
with  the  solutions  of  problems,  including  the  solutions  of  problems 
by  means  of  equations  containing  more  than  one  unknown  number. 

57.  Degree  of  an  Equation.  — The  degree  of  a  term  is  the  sum  of 

the  exponents  of  its  literal  factors.     If  there  is  only  one  literal 
factor,  its  exponent  is  the  degree  of  the  term. 

Per  example :  aj  is  of  the  Jirst  degree^ 

5  x'-^  is  of  the  second  degree^ 
2  a%  is  of  the  third  degree., 
3  vfiv^  is  of  the  fourth  degree. 

The  degree  is  expressed  sometimes  with  respect  to  some  one 
letter. 

Thus,  4  AnH^  is  of  the  first  degree  with  respect  to  ^,  of  the  second  degree 
with  respect  to  M,'and  of  the  third  degree  with  respect  to  t. 

The  degree  of  an  equation  is  the  degree  of  its  term  of  highest 
degree  with  respect  to  the  unknown  number  or  numbers. 

Thus,  w  +  3  =  4  w        is  of  the  first  degree., 

ic2  —  3  aj  =  8  ?/         is  of  the  second  degree., 
^^  +  i?2  _  6  —  0  is  of  the  third  degree., 

y^z'^  —  y^  =  2  2/  +  3  is  of  the  fourth  degree. 

An  equation  of  the  first  degree  is  called  a  linear  equation. 
One  of  the  second  degree  is  called  a  quadratic  equation. 
One  of  the  third  degree  is  called  a  cubic  equation. 


LINEAR  EQUATIONS:  PROBLEMS  93 

58.  Linear  Equations  with  One  Unknown  Number.  —  The  process 
of  solving  a  linear  equation  with  one  unknown  number  that  does 
not  contain  a  sign  of  grouping  was  given  in  §  15  and  §  36.  In  solv- 
ing linear  equations  that  contain  signs  of  grouping,  the  latter  should 
first  be  removed  by  use  of  the  principles  in  Chapters  IV  and  V. 

Example  1.— Solve     P-(6  -  2  P)  =  9(P- 1). 

Here  the  first  parentheses  are  removed  by  changing  the  signs  of  the  terms 
inclosed,  and  the  second  by  performing  the  indicated  multiplication,  giving  : 

P_6  +  2P=9P-9, 
p  +  2P-9P=6-9, 
-  6  P  =  -  3, 

P  =  h. 

Check.  —  When  P  —  \  the  equation  becomes 
^_(6_l)=  9(^-1), 
or  -4i  =  -4^. 

Example  2.  —  Solve  ( w  -  2)  (n  -  3)  =  (n  -  4)  (n  -  5) . 
The  parentheses  are  removed  by  multiplication,  giving : 
n^  -  5  n  +  6  =  n2  _  9  n  +  20, 
n2  -  n2  -  5  n  +  9  n  =  20  -  6, 
4  w  =  14, 
n  =  3J. 

Check.  —  When  n  =  3^  the  equation  becomes 

(3i-2)(3i-3)  =  (3J-4)(3i-5), 
or  f  =  |. 

EXERCISES 
Solve  and  check : 

1.  3(a  +  l)=12  +  4(a-l).       4.   5(4-3i?)  =  7(3- 4i2). 

2.  3(ar-2)  =  2(a;-3).  5.    2t-{ot  +  ^)  =  l. 

3.  5(2-42/)  =4(1 -32/).  6.   3(v  + 1) +5(v  -  1)  =  0. 

7.  3(d-14)  =  7(d-18). 

8.  2(Tr-l)+3(Tr-2)  +  4(Tr-3)=0. 

9.  2P-5(20-P)-6  =  0. 

10.   5(2/>4-l)-7  =  3(2i;-7)  +  51. 


94  ELEMENTARY  ALGEBRA 

11.  2(m-2)  +  3(m-3)  =  20-4(m-4). 

12.  5A;  +  6(A:  +  l)  =  7(A;  +  2)  +  8(A;H-3). 

13.  2c-3(c-4)+20  =  2c-hl7. 

14.  85-3(2F+7)  =  6F-+-4(4F+2). 

15.  4(5-f 4£')-5(6  +  4^)+100  =  2^  +  36. 

16.  12r-(4y-7)  =  3  T-(9r-28). 

17.  (5-n)(l  +  w)  =  (2-n)(4  +  n). 

18.  {K-l){K-2)  =  {K+^){K-4.). 

19.  (s  -  1)  (s  -  4)  =  2  s  +  (s  ~  2)  (s  -  3). 

20.  (5  r+  7)  (3  Y-  8)  =  (5  Y-\-  4)  (3  F-  5). 

21.  (4^-7)(4^-7)=(2^-5)(8^  +  3). 

22.  (5  -  3i0  (4  w  +  3)  -f- 1  =  (3  z^  +  7)(1  -4w). 

23.  (Q  +  4)(e  +  8)-e(Q  +  4)  =  128. 

24.  (4  -  3  ?/;)  (5  +  4  w)  =  (8  +  2  w;)  (1  -  6  w)  -  82. 

25.  (a  — 4)  (a +  4)  =  a^  — 8  a. 

26.  2(2;  +  2)(2-4)  =  2;(22;  +  l)-21. 

69.  Problems  solved  by  Linear  Equations.  —  The  following  ex- 
ample illustrates  a  type  of  problems  solved  by  means  of  linear 
equations  involving  signs  of  grouping. 

Example.  —  A  man  made  two  investments  in  railroad  stocks,  amounting 
together  to  $  15,000.  On  the  first  he  gained  14%,  and  on  the  second  he  lost 
5  %.  His  net  profits  from  the  two  investments  amounted  to  $  1055.  How 
many  dollars  in  each  investment  ? 

Let  a  =  amount  of  first  investment,  in  dollars. 

Then,  15000  —  a  =  amount  of  second  investment. 

.14  a  =  gain  on  first  investment. 
.05  (15000  —  a)  =  loss  on  second  investment. 
Hence,  .14  a  -  .05(15000  -  a)  =  1055. 
.14  a -750+  .05  a  =  1055. 
.19  a  =  1805. 
a  =  9500. 
15,000  -  a  =  5500. 

Therefore  the  first  investment  was  $  9500,  and  the  second,  1 5500. 


LINEAR  EQUATIONS:  PROBLEMS  95 

For   the    steps    in  the    process    of  expressing  a  problem   by 
raeans  of  an  equation  involving  one  unknown  letter,  as  illustrated 
n  the  above  example,  review  §  16. 

/ 
^         EXERCISES 

1.  The  sum  of  two  numbers  is  25,  and  twice  the  larger  exceeds 
four  times  the  smaller  by  2.     Find  the  numbers. 

2.  Separate  34  into  two  parts  such  that  twice  the  greater  shall 
be  less  by  12  than  5  times  the  smaller. 

3.  The  sum  of  two  numbers  is  64.  Three  times  the  less  is  12 
more  than  twice  the  greater.     Find  the  numbers. 

4.  Separate  86  into  two  parts  such  that  three  times  the  greater 
shall  exceed  5  times  the  smaller  by  2. 

5.  There  are  two  consecutive  odd  numbers  whose  product  ex- 
ceeds the  square  of  the  less  by  34.     Find  the  numbers. 

6.  When  the  square  of  a  whole  number  is  increased  by  29  the 
result  is  found  equal  to  the  product  of  the  two  next  larger  con- 
secutive whole  numbers.     Find  the  three  consecutive  numbers. 

(^  I  bought  i  lb.  of  coffee  and  5  lb.  of  tea  for  $5.40.  The  tea 
cost  45  cents  a  pound  more  than  the  coffee.  Find  the  cost  of  each 
per  pound. 

(^  A  newsboy  sold  51  papers  for  70  cents,  of  which  some  were 
one-cent  papers  and  the  others  two-cent  papers.  How  many 
papers  of  each  kind  were  there  ? 

9.  A  newsboy  sold  two-cent  evening  papers  and  Saturday 
Evening  Posts  (price  5  cents),  receiving  in  all  92  cents.  The 
total  number  of  papers  and  Posts  together  was  34.  How  many 
were  there  of  each  ? 

10.  Some  boys  had  a  refreshment  booth  at  which  they  sold 
lemonade  and  cider.  The  price  of  lemonade  was  2  cents  a  glass 
and  of  cider  5  cents  a  glass.  In  an  afternoon  they  sold  43  glasses, 
and  in  counting  their  money  found  that  they  had  $1.31.  How 
many  glasses  of  each  did  they  sell  ?  \ 


96  ELEMENTARY  ALGEBRA 

11.  Two  tanks  contained  equal  amounts  of  oil.  But  after  75 
gallons  had  been  taken  from  one,  and  50  gallons  added  to  the 
other,  one  contained  twice  as  much  as  the  other.  How  many 
gallons  did  each  contain  originally  ? 

12.  What  amount  must  be  subtracted  from  each  of  the  numbers 
12,  14,  18, 10,  so  that  the  product  of  the  first  two  remainders  shall 
equal  the  product  of  the  last  two  ? 

^^3.    Two  electric  lights  have  together  a  strength  of  128  candle- 
*^wer.     One  has  twice  as  much  and  16  candle-power  more  than 
the  other.     HoW  many  candle-power  has  each  ? 

14.  Two  grades  of  coffee  costing  the  dealer  28  cents  and  36 
cents  per  pound  are  to  be  mixed  so  that  the  mixture  shall  cost 
30  cents  per  pound.  What  parts  must  he  take  to  make  50  pounds 
of  the  mixture  ? 

15.  How  can  a  merchant  mix  10  pounds  of  tea,  one  kind  cost- 
ing 50  cents  and  the  other  65  cents  per  pound,  so  that  the  mixture 
shall  cost  60  cents  per  pound  ? 

16.  Two  grades  of  spice  worth  25  cents  and  45  cents  a  pound 
are  to  be  mixed  so  that  the  mixture  can  be  sold  for  50  cents  a 
pound  and  at  a  profit  of  25  %.  What  parts  must  be  taken  to 
make  8  pounds  of  the  mixture  ? 

17.  A  grocer  received  a  shipment  of  80  dozen  eggs.  Part  of 
these  were  sold  at  28  cents  per  dozen  and  the  rest  at  30  cents  per 
dozen.  The  total  receipts  from  the  sale  of  the  eggs  amounted  to 
$  23.12.     How  many  dozens  were  sold  at  each  price  ? 

18.  Listerine  contains  25  %  alcohol.  When  used  to  spray  the 
throat,  it  should  be  diluted  by  adding  water  to  it.  How  much 
water  must  be  added  to  100  parts  of  listerine  so  that  the  mixture 
contains  only  15  %  alcohol  ? 

19.  If  a  medicine  contains  40  %  alcohol,  how  much  of  other  in- 
gredients must  be  added  to  10  quarts  of  it  so  that  the  mixture 
shall  contain  only  25  %  alcohol  ? 

20.  How  many  quarts  of  water  must  be  mixed  with  40  quarts 
of  alcohol,  80  %  pure,  to  make  a  mixture  75  %  pure  ? 


LINEAR  EQUATIONS:  PROBLEMS  97 

21.  How  many  quarts  of  water  must  be  added  to  60  quarts  of 
alcohol,  85  %  pure,  to  make  a  mixture  75  %  pure  ? 

22.  How  many  gallons  of  cream  containing  28  %  butter  fat  and 
milk  containing  4  %  butter  fat  must  be  mixed  to  make  10  gallons 
of  cream  containing  25  %  butter  fat  ? 

23.  How  many  gallons  of  cream  containing  30  %  butter  fat  and 
milk  containing  4.5  %  butter  fat  must  be  mixed  to  make  20  gallons 
of  cream  containing  20  %  butter  fat  ? 

24.  In  an  alloy  of  gold  and  silver  weighing  60  ounces,  there  are 
6  ounces  of  gold.  How  much  silver  must  be  added  in  order  that 
10  ounces  of  the  new  alloy  shall  contain  only  |^  ounce  of  gold  ? 

25.  In  an  alloy  of  copper  and  tin  weighing  36  ounces,  there  are 
12  ounces  of  copper.  How  many  ounces  of  copper  must  be  added 
to  this  in  order  that  the  new  alloy  may  have  4  ounces  of  copper  to 
every  10  ounces  of  the  alloy  ? 

26.  A  man  has  two  investments  together  amounting  to  $  1800. 
On  the  first  investment  he  gets  5  %  annually,  and  on  the  other  he 
gets  6  (fo  annually.  His  annual  income  from  the  two  investments 
is  $  95.     Find  the  amount  of  each  investment. 

27.  $5000  is  invested  in  two  places,  part  at  4  %  and  the  rest 
at  5  %.  The  annual  income  from  the  two  investments  is  $222. 
How  much  is  each  investment  ? 

28.  One  sum  of  money  exceeds  another  by  $  1400.  The  first 
at  8  %  and  the  second  at  6  %  give  an  annual  income  of  $  280. 
Find  the  value  of  each. 

29.  There  are  51  coins  in  a  money  drawer,  consisting  of  nickels 
and  dimes.  The  total  value  is  $3.35.  How  many  coins  of  each 
kind  are  there  ? 

30.  The  value  of  29  coins,  consisting  of  quarters  and  dimes,  is 
$4.55.     Find  the  number  of  each. 

31.  A  is  42  years  of  age,  and  B  is  12.  In  how  many  years  will 
A  be  only  twice  as  old  as  B  ? 


98  ELEMENTARY  ALGEBRA 

32.  Eight  years  ago  a  man  was  just  16  times  as  old  as  his  son, 
and  now  he  is  only  4  times  as  old.     What  are  their  present  ages? 

33.  Find  where  to  cut  a  board  which  is  50  inches  long  into  two 
parts  whose  difference  is  26  inches. 

34.  A  bar  of  iron  60  inches  long  is  to  be  cut  into  two  parts  such 
that  twice  one  part  is  equal  to  5  times  the  other.  Find  where  to 
cut  it. 

35.  A  belt  runs  over  a  pulley  48  inches  in  diameter,  making 
180  revolutions  a  minute.  It  is  desired  to  reduce  the  size  of  the 
pulley  so  that  by  making  216  revolutions  a  minute,  the  belt  will 
move  with  the  same  speed.  By  how  much  must  the  diameter  of 
the  pulley  be  reduced  ? 

36.  A  rectangular  fiel^  is  6  rods  longer  than  it  is  wide;  and  if 
the  length  and  breadth  were  each  4  rods  more,  the  area  would  be 
120  square  rods  more  than  it  is.  What  are  the  dimensions  of  the 
field  ? 

37.  A  square  court  has  the  same  area  as  a  rectangular  court 
whose  length  is  18  feet  greater  and  width  9  feet  less  than  the  side 
of  the  square  court.     Find  the  side  of  the  square  court. 

38.  A  tennis  court  is  6  feet  more  than  2  times  as  long  as  it  is 
wide,  and  its  area  exceeds  by  216  square  feet  twice  the  area  of  the 
square  whose  side  is  its  width.     Find  the  dimensions  of  the  court. 

60.  Business  Problems.  —  Certain  kinds  of  business  problems 
are  easily  solved  by  use  of  linear  equations. 

EXERCISES 

1.  A  university  has  an  endowment  which  invested  at  4  % 
yields  an  annual  income  of  $  140,000.     What  is  the  endowment  ? 

Suggestion.  —  If  d  dollars  is  the  endowment,  .04(?  =  140,000. 

2.  What   sum   of   money   invested  at  6  %    will  yield  $510 
simple  interest  per  year  ? 

3.  A  certain  investment  at  4}  %  simple  interest  yields  $  3240 
in  6  years.     What  is  the  investment  ? 


LINEAR    EQUATIONS:   PROBI.EMS  i  i  ■  "•  '  J  :  s^dV, 

4.  A  certain  sum  of  money  was  invested  at  5  %  simple  interest. 
In  3  years  the  principal  and  interest  together  amounted  to  $  6900. 
What  was  the  amount  of  the  investment  ? 

5.  Five  years  ago  I  invested  a  certain  sum  of  money  at  6  % 
simple  interest.  It  now  amounts  to  $1560.  How  much  did  I 
invest  ? 

6.  A  number  of  years  ago  I  invested  $  1500  at  5  %  simple  in- 
terest. The  amount  at  present  is  $2025.  How  many  years  ago 
was  the  investment  made  ? 

7.  A  commission  merchant  charged  $  10.80  for  selling  a  car  load 
of  fruit,  and  remitted  $  349.20.    What  was  his  rate  of  commission  ? 

8.  Merchants  sometimes  mark  goods  to  sell  at  an  advance,  and 
then  allow  a  discount  on  the  marked  price.  At  what  advance 
must  a  merchant  mark  goods  costing  $12  in  order  that  he  may 
sell  them  at  a  reduction  of  20  %  from  the  marked  price,  and  yet 
make  a  profit  of  25  %  on  the  goods  ? 

SuGGESTiov.  —  If  a  =  the  advance,  in  dollars,  the  marked  price  will  be 
12  +  a,  and  the  selling  price  80  %  of  this,  or  .80(12  +  a).  Hence,  .80(r2  +  a) 
=  12  +  .25  X  12. 

9.  At  what  advance  must  a  merchant  mark  an  article  costing 
him  $3.50  in  order  that  he  may  sell  it  at  a  reduction  of  10  %  from 
the  marked  price,  and  yet  make  a  profit  of  30  %  on  it  ? 

10.  A  clothing  merchant  puts  on  sale  a  lot  of  boys'  suits  costing 
him  $8  each,  and  advertises  them  to  sell  at  a  reduction  of  25  % 
from  the  marked  price.  At  what  advance  must  he  first  mark 
them  to  sell  in  order  that  he  may  make  this  reduction  and  still 
make  a  profit  of  20%  on  the  cost  of  the  suits  ? 

11.  A  milliner  advertises  a  lot  of  hats  that  cost  her  $  2.85  each 
for  special  sale  at  "  ^  off."  At  what  advance  must  she  first  mark 
them  in  order  that  by  allowing  this  discount  she  may  make  a  profit 
of  75  cents  on  each  hat  ? 

12.  The  proprietor  of  a  china  store  marks  for  sale  a  lot  of  dishes 
that  cost  him  $  16.50.  At  what  advance  must  he  mark  them  so 
that  he  may  then  mark  them  down  15  %  and  make  a  profit  of  20  %  ? 


if)>J  :'\'\\\      \  /.mCEMENTARY  ALGEBRA 

13.  A  ladies'  tailoring  establishment  marked  a  suit  at  an 
advance  of  50%,  then  reduced  the  selling  price  by  20%  of  the 
marked  price,  and  got  $16.80  for  it.  What  was  the  cost  of 
the  suit? 

14.  A  druggist  had  marked  perfume  to  sell  at  a  profit  of  40  % . 
He  then  advertised  it  for  special  sale  at  42  cents  an  ounce,  which 
was  a  reduction  of  14f  %  from  the  marked  price.  What  did  it 
cost  the  druggist  per  ounce  ? 

15.  Men's  suits  that  had  been  marked  to  sell  at  a  gain  of  25  % 
were  damaged  and  disposed  of  at  a  fire  sale  at  half  price,  which 
was  a  loss  of  $  7.50  on  each  suit.     What  was  the  cost  of  the  suits  ? 

61.  Problems  involving  Motion. — When  an  object  moves,  the 
distance  that  it  goes  in  a  unit  of  time,  as  a  second  or  an  hour,  is 
called  its  rate  of  motion,  speed,  or  velocity. 

For  example :  If  a  train  runs  60  miles  in  2  hours,  its  velocity  is  30  miles 
an  hour. 

If  the  velocity  does   not  change   throughout  the  motion,  the 

motion  is  said  to  be  uniform.     In  case  a  body  moves  with  uniform 

motion,  the  distance  that  it  goes  in  any  length  of  time  is  obtained 

by  multiplying  the  velocity  by  the  number  of  units  of  time ;  that 

is, 

distance  =  time  x  velocity. 

EXERCISES 

1.  An  express  train  runs  with  a  velocity  of  45  miles  an  hour. 
How  far  does  it  go  in  4  hours  ? 

2.  The  "  Kocky  Mountain  Limited,"  on  the  Kock  Island  Koad, 
runs  from  Omaha  to  Denver,  a  distance  of  580  miles,  in  14  hours 
and  48  minutes.  What  is  the  average  speed  of  the  train,  making 
no  allowances  for  stops  ? 

3.  Sound  travels  through  the  air  1080  feet  a  second.  If  a 
flash  of  lightning  is  a  mile  away,  how  long  after  the  flash  should 
one  hear  the  thunder  ? 


LINEAR  EQUATIONS:  PROBLJ^MS  iOl 

4.  A  train  leaves  a  station  and  runs  at  a  speed  of  40  miles  an 
hour.  Two  hours  later  a  second  train  leaves  the  same  station 
and  runs  over  the  same  track  at  the  rate  of  55  miles  an  hour. 
In  how  many  hours  will  the  second  train  pass  the  first  ? 

Suggestion. —  Let  t  =  number  of  hours  after  second  train  starts  until  it 
passes  the  first.  Then,  since  the  first  train  has  a  start  of  two  hours, 
55«  =  40(e  +  2). 

5.  A  messenger,  going  6  miles  per  hour,  has  been  gone  for 
two  hours  when  it  is  found  that  the  message  is  wrong.  A  second 
messenger,  riding  at  an  average  of  10  miles  per  hour,  is  sent  to 
overtake  him.  In  how  many  hours  will  the  second  messenger 
overtake  the  first  ? 

6.  Two  motor  cyclists  start  at  the  same  time  from  points  280 
miles  apart,  and  travel  towards  each  other.  One  rides  24  miles 
per  hour,  but  is  delayed  3  hours  on  the  road  because  of  a  break 
in  his  machine.  The  other  rides  20  miles  per  hour  uninter- 
ruptedly.    In  how  many  hours  will  they  meet  ? 

7.  Two  pedestrians  started  at  the  same  time  from  points  44| 
miles  apart,  one  walking  at  the  rate  of  2^  miles  an  hour,  and  the 
other  at  the  rate  of  2|  miles  an  hour.  When  and  where  did  they 
meet? 

8.  Two  automobile  parties  started  from  the  same  place,  one 
going  north  at  20  miles  an  hour  and  the  other  south  at  18  miles 
an  hour.     In  what  time  will  they  be  120  miles  apart  ? 

9.  A  motor  cyclist  rode  75  miles  in  4  hours.  Part  of  the  dis- 
tance was  on  a  country  road  at  a  speed  of  20  miles  an  hour,  and 
the  rest  within  the  city  limits  at  10  miles  an  hour.  Find  how 
many  hours  of  his  ride  were  in  the  country. 

10.  A  train  runs  from  Joliet,  111.,  to  Chicago,  39.6  miles,  in  1.1 
hours.  The  run  from  Joliet  to  Englewood  is  made  at  40  miles  an 
hour,  and  the  run  from  Englewood  to  Chicago  at  24  miles  an 
hour.  Find  the  time  required  to  make  the  run  from  Englewood 
to  Chicago.     Find  the  distance  from  Englewood  tO  Chicago. 


^Wa*.* -•'.••;;    :  cEEe^mentary  algebra 
**  i  ,*'**  *t  ,"  *  *  •     *  •*, « 

11.  A  train  running  from  Chicago  to  Denver  at  the  average 
speed  of  40  miles  an  hour  takes  three  hours  longer  to  make  the 
run  than  one  running  at  45  miles  an  hour.  Find  the  distance 
from  Chicago  to  Denver. 

Suggestion.  —  First  find  the  time  required  at  a  speed  of  40  miles  an  hour. 

12.  An  ocean  liner  going  20  knots  an  hour  leaves  New  York 
when  a  freighter  going  6  knots  an  hour  is  already  90  knots  out. 
How  long  will  it  take  the  liner  to  overtake  the  freighter  ? 

13.  A  travels  8  hours  at  a  rate  of  3  miles  per  hour  less  than 
the  rate  of  B.  B  travels  an  equal  distance  in  6  hours.  What  is 
the  rate  of  each  ? 

14.  Two  trains  approach  each  other,  leaving  stations  149  miles 
apart  at  the  same  time.  One  goes  10  miles  per  hour  faster  than 
the  other,  and  they  meet  in  2  hours.     What  is  the  rate  of  each  ? 

15.  The  longest  steel  bridge  in  the  world  is  said  to  be  one  of 
the  Spokane,  Portland,  &  Seattle  E-ailroad,  over  the  Columbia 
River.  Including  approaches,  it  is  2  miles  long.  It  would  take 
a  freight  train  1144  feet  long,  running  15  miles  per  hour  (1320 
feet  per  minute),  just  3  minutes  to  cross  the  ten  main  spans. 
Find  the  length  of  the  ten  main  spans. 

16.  A  man  shooting  at  a  target  heard  the  bullet  strike  the  tar- 
get 3|  seconds  after  he  fired.  The  bullet  was  known  to  travel 
1375  feet  a  second ;  and  sound  travels  approximately  1100  feet  a 
second.  How  long  after  he  fired  did  the  bullet  hit  the  target  ? 
How  far  away  was  the  target  ? 

17.  A  bullet  going  1650  feet  per  second  is  heard  to  strike  a  tar- 
get 2  seconds  after  it  is  fired.  How  long  after  it  is  fired  does  it 
hit  the  target  ?     How  far  away  is  the  target  ? 

18.  Find  the  time  between  4  and  5  o'clock  when  the  hands  of 
a  clock  are  together. 

Suggestion.  —  Let  x  =  the  number  of  minute  spaces  which  the  minute 
hand  has  traveled  from  4  o'clock  on  until  it  overtook  the  hour  hand.  Then 
■^  X  will  be  the  number  of  minute  spaces  which  the  hour  hand  has  traveled 
meanwhile.     Why  ?    The  difference  is  20.     Why  ? 


LINEAR   EQUATIONS:  PROBLEMS 


103 


19.  Find  the  time  between  7  and  8  o'clock  when  the  hands  of 
a  clock  are  together. 

20.  At  what  time  between  6  and  7  o'clock  is  the  minnte  hand 
15  minute  spaces  behind  the  hour  hand  ? 

Suggestion.  —  Starting  at  6  o'clock,  how 
many  minute  spaces  has  the  minute  hand  had 
to  gain  on  the  hour  hand  ? 

21.  Find  the  time  between  4  and  5 
o'clock  when  the  hands  of  a  clock  are 
directly  opposite  each  other. 

22.  At  what  time  between  8  and  9 
o'clock  are  the  hands  of  a  clock  directly 
opposite  each  other  ? 

23.  How  long  is  it  from  the  time  that  the  hour  and  minute 
hands  of  a  clock  are  together  until  they  are  together  again  ? 

Suggestion.  —  Let  t  =  the  number  of  minutes  required.    Since  the  minute 
hand  makes  one  revolution  in  60  minutes,  it  makes  ^^  of  a  revolution  in  one 

minute,  and  ^  t  revolutions  in  t  minutes.  Simi- 
larly, the  hour  hand  makes  7^5  of  a  revolution 
in  one  minute,  and  ^q  t  revolutions  in  t  min- 
utes.    Hence,  ^  <  —  ^i©  <  =  1. 


24.  The  earth  makes  a  circuit  around 
the  sun  in  12  months,  and  Mercury 
makes  a  circuit  around  the  sun  in  3 
months.  If  the  earth  and  Mercury  are 
"  in  conjunction,"  as  in  the  figure,  how 
long  will  it  be  until  they  are  in  conjunction  again? 

25.  Venus  makes  a  circuit  around  the  sun  in  225  days,  and 
Mars  in  687  days.  How  many  days  elapse  between  two  consecu- 
tive conjunctions  of  these  two  planets  ? 

62.  Problems  on  Specific  Gravities  of  Substances.  —  A  cubic  foot 
of  steel  weighs  7.8  times  as  much  as  a  cubic  foot  of  water.  This 
number,  7.8,  is  called  the  specific  gravity  of  steel.     In  general, 


104  ELEMENTARY  ALGEBRA 

The  ratio  of  the  weight  of  a  given  volume  of  any  solid  or  liquid 
substance  to  the  weight  of  a7i  equal  volume  of  water  at  the  freezing 
point  of  temperature  is  the  specific  gravity  of  the  substance. 

What  does  it  mean  to  say  that  the  specific  gravity  of  alcohol  is 
.79? 

A  cubic  centimeter  of  distilled  water  at  the  freezing  point 
weighs  just  1  gram.  Since  the  specific  gravity  of  pure  gold  is 
19.36,  1  cubic  centimeter  of  gold  weighs  19.36  grams;  2  cubic 
centimeters  weigh  2  x  19.36  grams ;  3  cubic  centimeters  weigh 
3  X  19.36  grams ;  etc.     Evidently, 

The  weight  of  an  object  in  grams  equals  the  product  of  its  volume 
in  cubic  centimeters  and  its  specific  gravity. 

Thus,  if  the  specific  gravity  of  copper  is  8.9,  1  cubic  centimeter  of  cop- 
per weighs  8.9  grams.  Hence,  10  cubic  centimeters  of  copper  weigh 
10  X  8.9  grams,  or  89  grams. 

Certain  problems  in  the  specific  gravities  of  mixtures  of  sub- 
stances are  solved  by  use  of  linear  equations. 

EXERCISES 

1.  The  specific  gravity  of  cast  iron  is  7.2.  Find  the  weight 
of  10  cubic  centimeters  of  cast  iron.     Of  200  cubic  centimeters. 

2.  The  specific  gravity  of  glass  is  2.89.  Find  the  weight  of 
100  cubic  centimeters  of  glass.     Of  25  cubic  centimeters. 

3.  Brass  is  made  of  copper  and  zinc,  and  its  specific  gravity 
is  8.4.  How  many  cubic  centimeters  of  copper,  of  which  the 
specific  gravity  is  8.9,  must  be  used  with  200  cubic  centimeters 
of  zinc,  of  which  the  specific  gravity  is  6.9,  to  make  brass  ? 

Suggestion.  —  The  volume  of  the  brass  is  the  sum  of  tlie  volumes  of  the 
copper  and  zinc,  and  the  weight  of  the  brass  is  the  sum  of  the  weights  of  the 
copper  and  zinc.  Let  v  =  volume  of  copper  in  cubic  centimeters.  Then, 
weight  of  copper  =  8.9  u,  weight  of  zinc  =  6.9  x  200,  and  weight  of  brass 
=  8.4(i?  +  200).    Hence,  8.9 tj  +6.9  x  200  =  8.4(u  +  200). 

4.  How  much  zinc  must  be  combined  with  250  cubic  centi« 
meters  of  copper  to  form  brass  ? 


LINEAR   EQUATIONS:  PROBLEMS  105 

6.  How  many  cubic  centimeters  of  water  (specific  gravity  1) 
must  be  mixed  with  500  cubic  centimeters  of  alcohol  (specific 
gravity  .79)  so  that  the  specific  gravity  of  the  mixture  shall 
be  .9? 

6.  A  piece  of  ice  containing  1,000,000  cubic  centimeters,  spe- 
cific gravity  .92,  floats  in  water.  How  many  cubic  centimeters 
in  an  oak  beam,  specific  gravity  1.17,  that  may  be  placed  upon 
the  ice  without  making  it  sink  ? 

Suggestion. — The  specific  gravity  of  the  combination  of  ice  and  oak 
must  equal  1,  the  specific  gravity  of  water. 

7.  How  much  steel,  specific  gravity  7.8,  must  be  attached  to 
a  piece  of  white  pine,  specific  gravity  .42,  containing  10,000  cubic 
centimeters,  in  order  that  the  specific  gravity  of  the  combined 
materials  may  be  2  ? 

8.  In  Problem  7,  how  much  steel  must  be  used  so  that  the 
combined  materials  will  just  float  in  water? 

9.  A  piece  of  glass  containing  850  cubic  centimeters,  specific 
gravity  2.89,  is  made  to  float  by  attaching  cork  to  it.  How  much 
cork,  specific  gravity  .24,  must  be  used  ? 

10.  My  watch  case  is  made  of  14-karat  gold,  specific  gravity 
14.88.  How  much  pure  gold,  specific  gravity  19.36,  must  be 
combined  with  20  cubic  centimeters  of  nickel,  specific  gravity 
8.57,  to  make  14-karat  gold  ? 

11.  When  100  cubic  centimeters  of  mercury  and  10  cubic  centi- 
meters of  gold  (specific  gravity  19.36)  are  combined,  it  is  found 
that  the  specific  gravity  of  the  combined  materials  is  14.1.  Find 
the  specific  gravity  of  mercury. 

12.  It  is  found  that  when  90  cubic  centimeters  of  copper  (spe- 
cific gravity  8.9)  and  150  cubic  centimeters  of  tin  are  combined, 
the  specific  gravity  of  the  combination  is  7.9.  What  is  the  spe- 
cific  gravity  of  tin  ? 


106  ELEMENTARY  ALGEBRA 

63.    Problems  involving  the  Lever.  —  In  every  form  of  lever,  such 
as   the   steelyard,  crowbar,  nutcracker,  teeter   board,  etc.,  it   is 

found  that  the  two  weights  or  forces  ex- 

fj        Tg!^  erted  balance  when  they  are  placed  at 

'         ^■^"'J_g)         ^^^Y\  distances  from  the  point  of  ^\xi)- 
Y         port  or  fulcrum  that  the  products  of  the 
|w|     1     weights  or  forces  by  their  distances  from 
the  fulcrum  are  equal. 
Thus,  in  case  of  the  steelyard,  wd  =  WD. 

Some  problems  on  the  use  of  the  lever  may  be  solved  by  linear 
equations. 

EXERCISES 

1.  On  a  steelyard  the  distance  D  is  2  inches  and  the  weight  w 
is  ^  pound.     If  d  is  8  inches,  what  is  the  weight  W? 

2.  A  nutcracker  is  held  4  inches  from  the  hinge  or  fulcrum, 
and  a  nut  is  placed  1  inch  from  the  hinge. 
A  squeezing  force  of  3  pounds  is  required  to 
crack  the  nut.     What  is  its  resistance  ? 


3.  Two    children    play    at    teeter.      One 
weighs  80  pounds,  and  sits  5  feet  from  the 
point  of  support  of  the  teeter  board.     If  the 
board  balances  when  the  other  sits  6  feet  from  the  point  of  sup- 
port, what  is  the  weight  of  the  second  child  ? 

4.  Two  boys  play  at  teeter.  One  weighs  100  pounds,  and  sits 
6  feet  from  the  point  of  support.  The  other  weighs  120  pounds. 
How  far  from  the  point  of  support  must  he  sit  in  order  to  make 
the  board  balance  ? 

5.  A  board  12  feet  long  is  to  be  used  as  a  teeter  board.  If  the 
people  weigh  95  pounds  and  110  pounds,  respectively,  and  sit  at 
the  ends  of  the  board,  find  the  point  at  which  it  must  be  sup- 
ported in  order  to  balance. 

6.  A  workman  lifts  a  stone  weighing  420  pounds  by  means  of 
a  crowbar.     If  the  fulcrum  is  placed  6  inches  from  the  point  at 


EFNEAR  EQUATIONS:  PROBLEMS  107 

which  the  crowbar  is  applied  to  the  stone,  how  far  from  the  ful- 
crum must  the  workman  grasp  the  crowbar  to  lift  the  stone  with 
a  force  of  100  pounds  ? 

7.  In  Problem  6,  if  the  man  grasps  the       „^ 
crowbar  at  a  point  48  inches  from  the 
point  where  the  crowbar   is   applied   to 

the  stone,  where  must  the  fulcrum  be  placed  in  order  that  he  may 
lift  the  stone  with  a  pressure  of  70  pounds  ? 

8.  Weights  of  240  pounds  and  300  pounds,  respectively,  are 
applied  at  the  ends  of  a  beam  40  inches  long.  At  what  point 
must  the  beam  be  supported  in  order  to  balance  ? 

9.  A  and  B  carry  an  object  suspended  from  a  horizontal  rod 
60  inches  long  held  between  them.  If  one  is  to  lift  only  three 
fourths  as  much  as  the  other,  at  what  point  of  the  rod  must  the 
object  be  suspended  ? 

10.   A  man  has  a  team  of  which  one  horse  weighs  1200  pounds 

and  the  other  1500  pounds.  If 
their  draft  power  is  proportional  to 
their  weight,  how  must  he  divide 
the  50-inch  doubletree  in  order  to 
justly  distribute  the  load  ? 

11.  A  farmer  has  a  team  of  which 
one  horse  weighs  1400  pounds  and 
the  other  1600  pounds.  If  their  draft  power  is  proportional 
to  their  weight,  where  must  he  place  the  clevis  on  the  4-foot 
doubletree  so  as  to  justly  distribute  the  load  ? 

64.  Problems  on  the  Decimal  Number  System. — In  any  number 
such  as  4258,  the  8  is  called  the  ones'  digits  the  5  the  tens'  digit, 
and  2  the  hundreds'  digit,  etc. 

Any  number  of  two  or  more  digits  in  our  decimal  system  of 
notation  is  in  nature  a  polynomial.  Thus,  483  =  400  +  80  +  3,  or 
100  X  4  + 10  X  8  +  3.  Similarly,  any  number  whose  hundreds', 
tens',  and  ones'  digits  are  respectively  A,  t^  and  w,  may  be  written 


108  ELEMENTARY  ALGEBRA 

in  the  polynomial  form  100  h -[- 10 1  -\-  u.  In  writing  a  number  in 
the  polynomial  form,  each  digit  must  be  multiplied  by  10,  100, 
1000,  etc.,  according  to  the  position  that  it  occupies  in  the 
number. 

EXERCISES 

1.  Write  in  the  polynomial  form  the  following  numbers :  25; 
347;  4196. 

2.  Write  the  number  whose  ones'  digit  is  a,  tens'  digit  6,  hun- 
dreds' digit  c,  and  thousands'  digit  d. 

3.  In  a  number  of  two  digits,  the  sum  of  the  digits  is  9.  If 
the  digits  are  interchanged  in  position,  the  new  number  obtained 
exceeds  the  given  number  by  27.     Find  the  number. 

Suggestion.  —  If  a;  =  the  tens'  digit,  9  —  a;  =  the  ones'  digit. 
Hence,       10a;4-9  —  ic=:  the  number. 

Hence,     10(9  —  a;)  +  x  =  the  number  with  the  digits  interchanged. 
Hence,     10(9  -  a:)+ x  =  lOx  +  9  -  a;  +  27. 

4.  A  number  is  composed  of  two  digits  whose  sum  is  12.  This 
number  exceeds  by  36  the  number  obtained  by  interchanging  the 
digits.     Find  the  number. 

5.  In  a  certain  number  of  two  digits,  the  tens'  digit  exceeds  the 
ones'  digit  by  2.  The  sum  of  the  given  number  and  that  obtained 
by  interchanging  the  digits  is  154.     What  is  the  number? 

6.  A  number  consists  of  two  digits  of  which  the  ones'  digit  is 
3  more  than  the  tens'  digit,  and  the  number  is  3  more  than  4 
times  the  sum  of  its  digits.     What  is  the  number  ? 

7.  What  is  the  number  of  which  the  tens'  digit  is  2  more  than 
the  ones'  digit,  and  which  is  7  times  the  sum  of  the  digits  ? 

8.  In  a  number  of  two  digits  the  ones'  digit  is  3  less  than  the 
tens'  digit,  and  the  number  is  20  less  than  twice  that  obtained 
by  interchanging  the  digits.     What  is  the  number  ?. 

9.  In  a  number  of  three  digits^  the  tens'  digit  is  1  more  than 
the  ones'  digit,  and  the .  hundreds'  digit  is  1  more  than  the  tens' 
digit.  The  number  is  21  more  than  50  times  the  sum  of  its 
digits.     Find  the  number. 


LINEAR   EQUATIONS:  PROBLEMS  109 

65.  Linear  Equations  with  Two  Unknown  Numbers.  — Problems  in 
which  the  values  of  more  than  one  unknown  quantity  are  to  be 
found  are  often  most  easily  solved  by  use  of  equations  containing 
more  than  one  unknown  number. 

Example. — A  merchant  sold   10  suits  for  $162.  He  received  |15  for 

each  of  one  kind  and  $  18  for  each  of  the  other  kind.  How  many  were  there 
of  each  kind  ? 

Let  n  =  the  number  sold  at  1 15  each,  and  w  =  the  number  sold  at  $  18 
each. 

Then                                     \           n  +  m  =  \Q,  (1) 

1 15  n  +  18  w  =  162.      •  (2) 

Multiplying  (1)  by  15,          15  » +  15  m  =  150.  (3) 

Subtractmg  (3)  from  (2),                  3  w  =  12.  (4) 

Hence,                                                  w  =  4.  (5) 

Replacing  m  by  4  in  (1),                 n  +  4  =  10.  (6) 

Hence,                                                     n  =  6.  (7) 

Therefore,  6  suits  were  sold  at  $  16  each  and  4  suits  at  $  18  each. 

66.  Simultaneous  Equations.  — In  the  example  in  §  65,  we  have 
seen  two  equations  concerning  two  unknown  numbers  whose 
values  satisfy  both  equations. 

Thus,  in  the  example,  when  w=6  and  m=4,  equation  (1)  becomes 
6  +  4  =  10,  and  equation  (2)  becomes  90  +  72  =  162. 

Two  or  more  equations  containing  two  or  more  unknown  num- 
bers which  satisfy  all  of  the  equations  are  called  simultaneous. 

67.  Systems  of  Simultaneous  Equations.  —  Two  or  more  simul- 
taneous equations  are  said  to  constitute  a  system.  To  solve  a 
system  of  equations  is  to  find  the  sets  of  values  of  the  unknown 
numbers  which  satisfy  all  of  the  equations. 

Thus,  in  the  system  of  equations  in  §  65,  the  set  of  values  w  =  6  and  m  =4 
constitute  a  solution. 

68.  Elimination.  —  The  solution  of  a  system  of  two  linear 
equations  is  most  easily  found  by  so  combining  the  two  equations 
as  to  obtain  a  new  equation  in  which  one  of  the  unknown  num- 
bers does  not  appear.     This  process  is  called  elimination.     A  sim 


no  ELEMENTARY  ALGEBRA 

pie  method  of  elimination  is  shown  in  the  example  in  §  65. 
Compare  the  steps  in  the  process  of  solving  the  following  system 
with  those  in  the  example  in  §  65. 

Example.— Solve  the  system   i     ^+4^  =  12,  (1) 

l2^-    J5  =  6.  (2) 

Let  us  first  eliminate  B. 

Multiplying  (2)  by  4,  ^A-^B=  24.  (3) 

Adding  (3)  to  (1),  9  ^  =  36.  (4) 

Hence,  ^  =  4.  (5) 

The  value  of  B  may  now  be  found  in  like  manner  by  eliminating  A  be- 
tween equations  (1)  and  (2),  but  is  more  easily  found  by  replacing  A  by  its 
value  4  in  equation  (1)  or  equation  (2). 

From  (1),  when  ^  =  4,  4  +  4  5  =  12.  (6) 

Whence,  B  =  2.  (7) 

See  if  the  solution  A  =  4,  B  =  2  satisfies  both  equations  of  the  system. 

A  study  of  this  example  and  that  in  §  65  will  reveal  the  follow- 
ing rule : 

To  eliminate  one  of  the  unknown  numbers,  multiply  the  members 
of  each  equation,  if  necessary,  by  such  a  number  as  will  make  the 
numerical  coefficients  of  that  unknown  number  the  same  in  both  of 
the  resulting  equations.  Add  or  subtract  {according  to  signs)  the 
corresponding  members  of  the  resulting  equations. 

After  finding  the  value  of  one  unknown  number,  substitute  its  value 
in  either  of  the  given  equations  and  solve  the  resulting  equation  for 
the  value  of  the  other  unknown  number. 

Note.  —  Other  methods  of  elimination  will  be  discussed  in  Chapter  XII. 


EXERCISES 

Solve: 


\2n 
[3n 


|3a;_42/  =  6. 


2      f  5  Jf+4iV^=22,  g      f2n  +  5^  =  15, 


3Jf+    ^=9.  [3n-4AL=ll. 

{QE-    d  =  10,  r     P+4i?  =  12, 

\l  E-2d  =  \^.  '■     [2P-    E  =  6. 


LINEAR  EQUATIONS :  PROBLEMS  111 

3;2_c  =  8.  '     |5m-2n  =  14. 


9. 


{bt-2a: 
5T-4:H=2.  '"     [st  =  5a-ll. 

r5  Tr-2F-35  =  0,  j2      ri06?-llv  +  l 

|4  TF-    F-25  =  0.  ■     [10v-llG  = 


69.   Problems   solved  by  Use  of   Systems  of  Equations.  —  In  a 

problem  in  which  the  values  of  two  unknown  quantities  are 
sought  it  always  will  be  found  that  the  problem  expresses  or 
implies  two  facts  or  suppositions.  Represent  each  unknown  num- 
ber by  a  letter.  Then  each  of  the  facts  or  suppositions  expressed 
in  the  problem  gives  an  equation  containing  the  unknown  num- 
bers. These  two  equations  constitute  a  system,  which  may  be 
solved  by  the  method  of  §  68. 

Many  of  the  problems  in  the  exercises  in  the  earlier  part  of 
this  chapter  might  have  been  solved  by  use  of  two  equations  con- 
taining two  unknown  numbers. 

ExAiMPLE.  —I  have  $1.50  in  dimes  and  nickels.     There  are  25  coins  in  all. 
How  many  dimes  and  how  many  nickels  are  there  ? 
Let  d  =  number  of  dimes,  and  n  =  number  of  nickels. 

Then  I      ^+     *^  =  2^'  ^^) 

^       '  ll0d+5n  =  150.  (2) 

Note  that  the  fact  that  there  are  25  coins  in  all  gives  equation  (1),  and  the 
fact  that  their  total  value  is  $  1.50  gives  equation  (2). 
Solving  this  system,  ^f  =  5,  and  n  =  20. 
Hence,  there  are  5  dimes  and  20  nickels. 

EXERCISES 

Solve  by  using  two  equations  with  two -unknown  numbers: 
1.   A  man  invested  S5600,  part  in  bonds  that  paid  4%,  and 
the  remainder  in  a  business  enterprise  that  yielded  8%  on  the 
investment.     The  total  yearly  earnings  from  the  two  investments 
amounted  to  $328.     Find  the  amount  of  each  investment. 


112  ELEMENTARY  ALGEBRA 

Suggestion.  — Let  a  =  amount  of  investment  in  bonds,  and  b  =  amount  of 
investment  in  business  enterprise. 


„,  ,  a  +  6  =  5600, 

^^^"»  ^04  a +  .08  6  =  328. 


1.. 


2.  A  man  had  $2150  to  invest.  Part  of  this  sum  he  loaned 
at  6  %  interest,  and  the  remainder  he  invested  in  stock  of  a  build- 
ing and  loan  association  that  paid  7  %  dividends.  From  the  two 
investments  his  annual  income  was  $142.50.  What  was  the 
amount  of  each  investment  ? 

3.  A  money  drawer  contains  $3.50  in  dimes  and  nickels. 
There  are  50  coins  in  all.  How  many  coins  of  each  kind  are 
there  ? 

4.  In  playing  teeter,  two  boys  use  a  board  12  feet  long.  •  One 
boy  weighs  80  pounds  and  the  other  110  pounds.  At  what  point 
must  the  board  be  supported  to  balance  ? 

5.  A  teamster  has  a  team  of  which  one  horse  weighs  1250 
pounds  and  the  other  1500  pqunds.  Assuming  that  draft  power 
is  proportional  to  weight,  how  should  he  divide  the  50-inch  double- 
tree in  order  to  properly  distribute  the  load  ? 

6.  A  merchant  mixes  28-cent  coffee  and  36-cent  coffee  to  sell 
at  30  cents  a  pound.  What  quantities  of  the  two  grades  of  coffee 
should  he  take  to  make  40  pounds  of  the  mixture  ? 

7.  The  specific  gravity  of  copper  is  8.9,  that  of  zinc  6.9,  and 
that  of  brass  8.4.  Find  the  number  of  cubic  centimeters  of  copper 
a.nd  the  number  of  cubic  centimeters  of  zinc  that  must  be  com- 
bined to  produce  600  cubic  centimeters  of  brass. 

Suggestion.  —  Let  c  cubic  centimeters  copper  and  z  cubic  centimeters 
zinc  be  the  amounts  required. 

f  c  +  z  =  600, 

'  I  8.9c  +  6.dz  =  8.4  (c  +  z). 

8.  How  much  gold,  specific  gravity  19.36,  and  how  much 
nickel,  specific  gravity  8.54,  must  be  combined  to  make  10  cubic 
centimeters  of  an  alloy  of  which  the  specific  gravity  is  12  ? 


LINEAR  EQUATIONS:  PROBLEMS  113 

9.  In  a  number  of  two  digits,  the  sum  of  the  digits  is  10,  and 
when  the  digits  are  interchanged,  the  number  is  increased  by  36. 
Find  the  number. 

10.  A  belt  runs  over  two  pulleys.  The  circumference  of  one 
pulley  is  2  feet  more  than  the  circumference  of  the  other.  One 
pulley  makes  3  revolutions  while  the  other  makes  2.  Find  the 
circumference  of  each  pulley. 

11.  When  the  Panama  Canal  was  finished,  the  distance  from 
New  York  to  San  Francisco  by  boat  was  reduced  by  7796  miles. 
The  distance  by  the  old  route  exceeded  twice  the  distance  by  the 
Panama  route  by  2502  miles.  Find  the  distance  by  the  old  route 
and  the  distance  by  the  Panama  route. 

12.  The  distance  from  New  York  to  Yokohama,  Japan,  was 
reduced  by  2952  miles  when  the  Panama  Canal  was  completed. 
If  the  distance  by  the  old  route  were  twice  as  great,  and  the  dis- 
tance by  the  Panama  Canal  three  times  as  great,  the  latter  would 
exceed  the  former  by  4184  miles.  Find  the  distance  by  each 
route. 

13.  The  ship  Mauretania,  built  in  1907,  is  583  feet  longer  than 
the  ship  Britannica,  built  in  1840.  The  Mauretania  is  169  feet 
more  than  three  times  as  long  as  the  Britannica.  Find  the  length 
of  each. 

14.  In  1847  the  Deutschlandf  the  first  ship  of  the  Hamburg- 
American  Line,  arrived  in  New  York.  The  present  Deutschland 
carries  15,283  tons  more  than  the  first  Deutschlayid.  The  tonnage 
of  the  present  Deutschland  exceeds  by  226  tons  22  times  the 
tonnage  of  the  old  Deutschland.     Find  the  tonnage  of  each. 

15.  It  is  stated  that  the  daily  ration  for  a  laboring  man  should 
contain  4  ounces  each  of  fat  and  protein.  In  pork  the  per  cent 
of  fat  is  26  and  the  per  cent  of  protein  is  13  ;  and  in  beans  the  per 
cent  of  fat  is  2  and  the  per  cent  of  protein  is  22.  How  many 
ounces  each  of  pork  and  beans  are  required  to  make  a  ration  for 
one  day  ? 


114  ELEMENTARY  ALGEBRA 


SUPPLEMENTARY  EXERCISES 

Note.  —  As  we  have  seen,  many  practical  formulae  contain  two  or  more 

unknown  numbers.     It  often  is  found  necessary  to  solve  such  formulae  for 

one  of  the  unknown  numbers  in  terms  of  the  others.    Thus,  the  circumference 

of  a  circle  is  computed  by  the  formula  0  =  2  irR.     If  this  is  solved  for  i?  in 

C 

terms  of  C,  we  get  R  = Such  formulae  are  solved  by  the  same  processes 

27r 

as  equations  with  one  unknown  number. 

1.  In  the  simple  interest  formula,  I=prt,  solve  for  t  in  terms 
of  7,  p,  and  r. 

2.  The  distance  an  object  moves  in  t  seconds  at  v  feet  per 
second  is  s  =  tv.     Solve  for  t. 

3.  The  area  of  a  triangle  is  expressed  by  ^  =  ^  BH.  Solve 
iovH. 

4.  The  area  of  a  trapezoid  is  expressed  by  A  =  ^II(B-\-  B'). 
Solve  for  B'. 

5.  The  relation  between  the  readings  on  the  Fahrenheit  and 
Centigrade  thermometers  for  any  temperature  is  expressed  by 
(7=1  {F-  32).     Solve  for  F. 

6.  The  formula  for  finding  the  horse  power  of  a  steam  engine 

is  H.  P.  =  ^   !\  ,  where  _p  =  the  mean  effective  pressure  in  pounds 
33000 

per  square  inch,  I  =  the  length  of  stroke  in  feet,  a  =  area  of  piston 

in  square  inches,  n  =  twice  the  number  of  revolutions.     Solve  for 

Z,  the  length  of  stroke. 

7.  In  a  polygon  with  n  sides  the  sum  of  all  the  angles,  in 
degrees,  is  given  by  s  =  180  (n  —  2).     Solve  for  n. 

8.  Solve  2ax  —b  =  cy  for  x. 

9.  Solve  A(n  —  1)  +  ^  =  n  f or  ^. 

10.  Solve  (F-  S)(V-}-  2S)=V'-{-S'  for  F. 

11.  Solve  2(2^-1) +3  =  a(^  + 2)  for  f. 

12.  Solve  R{R  +  Jc)-k(l-Ji)  =  Ii'  +  l  for  B. 


LINEAR   EQUATIONS:  PROBLEMS  116 

In  each  of  the  following  systems  solve  for  x  and  y : 

^3      r8x+      2/  =  60a,  jg      r3aj  +  762/  =  166, 

\lx-10y=^   9a.  '     |2a;  +  562/=136. 

14      f    a;  4-    ?/ =  30  w,  -g      f  5maj  — 2ny=63mn, 

I  3  a*  —  2  y  =  25  n.  |  2  mx  +    n?/  =  18  m/i. 

17.  A  square  has  the  same  area  as  a  rectangle  whose  length  is 
3  inches  greater  and  width  2  inches  less  than  the  side  of  the 
square.     Find  the  dimensions  of  the  square  and  of  the  rectangle. 

18.  In  a  certain  family  each  daughter  has  as  many  sisters  as 
brothers;  but  each  son  has  twice  as  many  sisters  as  brothers. 
How  many  children  are  in  the  family  ? 

19.  The  leader  in  a  "  guessing  game"  tells  each  of  the  others  to 
add  6  years  to  his  age,  multiply  the  result  by  4,  subtract  24  from 
the  product,  and  to  the  remainder  add  his  age.  When  the  results 
are  announced,  the  leader  tells  at  once  the  age  of  each  individual. 
If  an  individual  gives  80  as  the  result,  what  is  his  age  ? 

'  20.  Find  the  longitude  of  a  place  in  the  Central  Time  belt  at 
which  it  is  observed  that  the  local  or  sun  time  is  15  minutes  faster 
than  standard  time. 

21.  Find  the  longitude  of  a  place  in  the  New  York  Time  belt 
whose  local  time  is  10  minutes  slower  than  standard  time. 

22.  A  crop  of  60  bushels  per  acre  of  corn  takes  from  one  acre 
l>f  soil  58  pounds  less  of  phosphoric  acid  than  of  nitrogen,  and 
ii3  pounds  less  of  potash  than  of  nitrogen.  Five  times  the  amount 
of  phosphoric  acid  exceeds  twice  the  amount  of  potash  by  8 
pounds.  How  many  pounds  of  each  of  these  fertilizers  are  taken 
from  an  acre  of  soil  ? 

23.  Fifteen  pounds  of  tin  weigh  13  pounds  in  water,  and  15 
pounds  of  zinc  weigh  13.5  pounds  in  water.  How  much  tin  and 
how  much  zinc  in  an  alloy  which  weighs  56  pounds  in  air  and  49 
pounds  in  water? 

Note. — The  method  given  in  this  problem  of  finding  the  proportional 
parts  of  metals  in  an  alloy  by  first  weighing  the  pure  metals  and  the  alloy 


116  ELEMENTARY  ALGEBRA 

each  in  air,  then  in  water,  is  said  to  have  been  first  discovered  and  used  by 
Archimedes,  about  220  b.c,  in  determining  for  King  Hieron  of  Syracuse 
whether  a  crown,  claimed  by  the  maker  to  be  pure  gold,  was  not  alloyed  with 
silver. 

24.  Two  seconds  after  a  marksman  fires  he  hears  the  bullet  hit 
the  target,  which  is  440  yards  distant.  If  sound  travels  through 
the  air  at  a  velocity  of  1100  feet  a  second,  find  the  average  veloc- 
ity of  the  bullet. 

25.  Two  weights  balance  when  one  is  12  inches  and  the  other 
15  inches  from  the  point  of  support.  If  the  first  is  replaced  by  a 
weight  6  pounds  greater,  the  second  must  be  moved  3  inches  far- 

^rom  the  point  of  support  to  balance  it.     Find  the  weights. 


A  crew  that  can  row  6  miles  an  hour  down  a  stream  can  row 
2  miles  an  hour  up  the  stream.  Find  the  sjjeed  of  the  current 
and  the  speed  at  which  the  crew  can  row  in  still  water. 

27.  A  local  train  820  feet  long  and  an  express  train  500  feet 
long  run  on  parallel  tracks.  When  running  in  the  same  direction 
it  requires  88  seconds  for  the  express  to  pass  the  local,  but 
when  running  in  opposite  directions  it  requires  only  17.6  seconds 
for  them  to  pass.     Find  the  rates  of  the  two  trains. 


CHAPTER   VII 
SPECIAL  PRODUCTS  AND    QUOTIENTS 

70.  Special  Rules.  —  The  products  or  quotients  of  expressions 
of  many  forms,  called  type  forms,  may  be  written  down  at  sight 
by  special  rules,  without  performing  in  detail  the  complete  pro- 
cesses of  multiplication  or  division.  By  discovering  and  learning 
these  rules,  much  time  and  labor  in  multiplications  and  divisions 
may  be  saved.  They  reveal  also  many  "  short  cuts  "  in  the  mul- 
tiplication of  arithmetical  numbers  that  are  valuable  to  know, 
some  of  the  most  important  of  which  are  given  in  this  chapter. 
The  need  for  use  of  these  special  rules  presents  itself  so  often  in 
algebra  that  they  should  be  thoroughly  mastered  before  proceed- 
ing farther. 

71.  Squares  and  Cubes  of  Monomials. —  What  is  the  product  of 
a^xa^?     Of  5  mhi'  x  5  mV  ?     Of  ( -  3  xy^^{-  3  xy^^)  ? 

Since  (7  My  means  7  M*x7  M\  what  is  its  value  ?  Find  the 
value  of  (9  P'(^y.     Of  (-  4  h^lc^f. 

From  these  examples,  the  following  rule  is  evident : 
To  square  a  monomial,  square  the  numerical  coefficient ,  and  mul- 
tiply each  exponent  by  2. 

Since  (2  ijt^f  means  2  ty^  X  2  w;^  X  2  w^,  what  is  its  value  ?  Find 
the  value  of  (3  HHy.     Of  (-  4  A^B^CTf. 

These  examples  reveal  the  following  rule  : 

To  cube  a  monomial,  cube  the  numerical  coefficient,  and  multiply 
each  exponent  by  3. 

Similar  rules  may  be  discovered  for  raising  monomials  to  the 
fourth  power,  the  fifth  power,  etc. 

117 


118  ELEMENTARY  ALGEBRA 

EXERCISES 
Give  at  sight  the  values  of  the  indicated  powers 


1. 

{aj. 

11. 

(Ly. 

21.  (xyy. 

31. 

(a^s^vy. 

2. 

(E^y. 

12. 

W. 

22.    (aby. 

32. 

(Sa^fy. 

3. 

(wy. 

13. 

(sy. 

23o    (yhy. 

33. 

(5  MNy. 

4. 

(j^y. 

14. 

(xy. 

24.    (r.W^ 

2 

34. 

(2  a'b'cy. 

5. 

{py. 

15. 

oy. 

25.    (W'gy. 

35. 

(8  w'%y. 

6. 

(ny. 

16. 

(Qy. 

26.  {Hyy. 

36. 

(11  Any. 

7. 

(B^y, 

17. 

(vy. 

27.    (a^^^)^ 

37. 

(IBxy'zy. 

8. 

(vr^ 

18. 

(Ny. 

28.  (vyy. 

38. 

(6  vHy. 

9. 

(apy. 

19. 

(py. 

29.    (A^ny. 

39. 

(9  S'Hy, 

10. 

{Ay, 

20. 

(by. 

30.    (w^ny. 

40. 

(16  mhiyy 

41. 

(SD^wy. 

48.    (- 

-SE'Hy. 

55. 

3  &Ty. 

42. 

(5  x*Ty. 

49.    (- 

-  2  a^'cy. 

56. 

4  HYvy. 

43. 

(2a^n'by. 

50.    (- 

-lOES'vy. 

57. 

2P'v'iiy. 

44. 

{^pH'yy. 

51.    (- 

-M^wy. 

58. 

3  Z2y^)l 

45. 

(6  V'n'gJ. 

52.    (- 

-  3  a'b'(^\ 

59. 

2  why. 

46. 

(-  2  Ay. 

53.    (- 

-x^yy. 

60. 

Say  {2  ay. 

47. 

(-3mW)2 

54.    (- 

-2  any 

61. 

(2 

ny(-sny. 

72.  Square  Roots  and  Cube  Roots  of  Monomials.  —  What  is  the 
number  which  squared  gives  9  ?  25?  a^? 

The  number  whose  square  is  a  given  number  is  called  the 
square  root  of  the  given  number. 

Thus,  since  7^  =  49,  the  square  root  of  49  is  7. 

Similarly,  the  number  whose  cube  is  a  given  number  is  called 
the  cube  root  of  the  given  number. 

Thus,  since  (2  ^2)3  =  8  v^,  the  cube  root  of  8  ??«  is  2  v^. 

The  numbers  whose  fourth  powers,  fifth  powers,  etc.,  are  a 
given  number  are  called  the  fourth  root,  fifth  root,  etc.,  of  the 
given  number. 


SPECIAL  PRODUCTS  AND   QUOTIENTS  119 

A  root  of  a  number  is  indicated  by  placing  before  it  tiie  sign  -^^ 
called  the  radical  sign.  Usually  a  vinculum  is  attached  to  the 
radical  sign,  to  show  how  far  its  effect  is  to  extend.  To  indicate 
what  root  it  is,  except  in  the  case  of  a  square  root,  a  little  num- 
ber called  the  index  is  written  above  the  radical  sign. 

For  example,  the  square  root  of  n  is  written  Vn  ;  the  cube  root  of  n  is 
written  Vn  ;  the  fourth  root  of  n  is  written  Vn;  etc.  In  the  case  of  square 
root  the  index  2  is  understood  and  left  off. 

In  most  simple  practical  work,  one  needs  only  a  knowledge  of 
square  roots  and  cube  roots. 

Since  (+3)2  =  9  and  (-3)2  =  9,  V9  is  either  +3  or  -3. 
Similarly,  since  (+6)2=36  and  (-6)2  =  36,  V36  is  either  +6  or 
—  6.     In  general. 

Every  positive  number  has  two  square  roots,  which  have  the  same 
absolute  value,  one  positive  and  the  other  negative. 

The  two  square  roots  of  a  positive  number  are  often  written 
together,  with  a  double  sign  ± . 

Thus,   \/8l  =  ±  9  ;  Vlil  =  ±  12. 

Since  no  positive  or  negative  numbers  squared  can  give  a  nega- 
tive number, 

A  negative  number  has  no  square  root  that  can  be  expressed  as  a 
positive  or  negative  number. 

Note.  —  What  to  do  in  a  problem  where  the  square  root  of  a  negative 
number  is  required  to  be  found  will  be  discussed  later. 

Since  (+2)^  =  8  and  (-2)3  =  -8,  ■v/8  =  2  and  V^  =  -2. 
In  general, 

Any  number  has  only  one  cube  root  that  can  be  expressed  as  a 
positive  or  negative  number.  This  cube  root  of  a  positive  number  is 
positive,  and  of  a  negative  number  it  is  negative. 

Note.  — It  will  be  shown  later  that  every  number  has,  in  addition  to  the 
one  positive  or  negative  cube  root,  two  other  cube  roots  that  cannot  be  ex- 
pressed as  simple  positive  or  negative  numbers.  In  this  chapter  only  the 
positive  or  negative  cube  root  of  a  number  will  be  considered. 


120  ELEMENTARY  ALGEBRA 

Since  the  square  root  of  a  monomial,  when  squared,  must  give 
that  monomial,  it  follows  from  §  71  that : 

To  find  the  square  roots  of  a  monomial,  take  the  square  roots  of  the 
numerical  coefficient  and  divide  the  exponent  of  each  letter  by  2. 

Similarly,  it  follows,  from  §  71,  that : 

To  find  the  cube  root  of  a  monomial,  take  the  cube   root  of  the 
numerical  coefficient,  and  divide  the  exponent  of  each  letter  by  3. 


Thus,  \/64  a«64  =  ±  8  a%^  ;  VlOO  x'^y^  =  ±  10  ot^y.     And  V8  u^t^  =  2  wH^ ; 
v'-27mi%6  =  _  3  „i4^2. 

EXERCISES 

Give  at  sight  the  square  roots  of; 

1.  1,  4,  16,  81,  121,  49,  144,  36. 

2.  a*,  TF«,  »«,  P^,  n"',  A^\  2/'"- 

3.  M'\  L^,  jS^%  k^',  9x%  16  a\ 

4.  49  JBi«,  25  v',  100  M^\  S6a''b%  81  a^y\ 

5.  16  TT^F^^  9 m'n%  25 dH\  121  hy,  225riV/. 

6.  144  a¥",  49  w^'d',  169  a^y^^  625  R^m'^ 

7.  196  a^o^V,  256  H^T^a^  441  ^"a^/,  400  m^w^. 

Give  at  sight  the  cube  roots  of : 

8.  1,  8,  27,  64,  125,  216,  512,  1000. 

9.  -  1,  -  8,  -  27,  -  64,  -  125,  -  1000. 

10.  A',  n^%  t%  W,  x'\  P^,  r^'. 

11.  v'\  y',  M^\  a^,  -D\  -z'\  -L\ 

12.  _  10^%  -  T^,  -  8  x%  -  27  b'',  -  125  P". 

13.  64  i?«,  27  vH\  -  8  m'^^  - 125  nV,  -  A^tH\ 

14.  -  pYr^,  216  M\i'',  -  27  aj^-^/V,  343  V^^K  -  8  i^3/)i2 


SPECIAL  PRODUCTS  AND   QUOTIENTS  121 

15.  The  area  of  a  square  is  49  d^.  What  is  the  length  of  one 
3ide? 

16.  The  base  and  altitude  of  a  right  tri- 
angle are  8  x  and  6  ic,  respectively.  Find 
the  length  of  the  hypotenuse. 

Suggestion.  —  The  square  of  the  hypotenuse 
equals  the  sum  of  the  squares  of  the  base  and  altitude. 

17.  The  volume  of  a  cube  is  64  E^.    Find  the  length  of  one  edge. 

18.  The  diameter  D,  in  inches,  necessary  of  a  steel  shaft,  upon 
which  are  fastened  pulleys  that  drive  machines  in  a  factory,  in 
order  to  impart  H  horse  power  to  the  machinery  when  making  N 

3/8O  X  H 
revolutions  a  minute,  is  computed  by  the  formula  D  =  y\ — . 

Find  the  diameter  of  the  shaft  necessary  to  impart  80  horse 
power  when  making  100  revolutions  a  minute. 

73.  Equations  solved  by  finding  Square  Roots.  —  Many  equations 
may  be  solved  by  extracting  roots.  The  following  illustrate  the 
solutions  of  equations  of  the  second  degree  by  finding  square  roots. 

Example  1.  — Solve  P^  =  9. 
Since  P2  =  9,  P  =  V9 

=  ±3. 

Example  2.  —  Solve  y'^-A^-0. 

Transposing,  y-  =  49. 

Hence,  y  =  \/49 

=  ±7. 

EXERCISES 
Solve: 

1.  a^  =  25.  6.  i2--64  =  0.  11.  3p--4S=0, 

2.  f  =  Sl.  7.  ^-121  =  0.  12.  7i>--28=0. 

3.  A^  =  S(j.  8.  F' -225  =  0.  13.  49  =  ^2. 

4.  7r^lOO.  9.  2cr  =  8.  14.  144  =  .S^. 

5.  2/' -16  =  0.       10.  ^>^f•'  =  45.  15.  75=  3  m 


122 


ELEMENTARY  ALGEBRA 


16.  The  time  required  for  an  object  starting  from  rest  to  fall  a 
given  distance  is  found  from  the  equation  16  t"^  =  s,  where  t  is  the 
time  in  seconds  and  s  is  the  distance  in  feet. 

Find  the  time  required  for  an  object  to  fall  64  feet. 

SuGGESTiox.  — 16^2=64.  Solve  for  t.  Has  the  negative  answer  any 
meaning  ? 

17.  In  the  following  table  are  given  the  distances  that  an  ob- 
ject falls.  Complete  the  table  by  finding,  by  the  formula  in 
Problem  16,  the  time  required  in  each  case. 


Distance 

16  Ft. 

144  Ft. 

576  Ft. 

1600  Ft. 

Time 

"~St 


18.  In  the  manufacture  of  lids  for  metal  boxes  and  cans,  a 
circular  piece  of  metal,  called  the  "blank,"  is  cut  from  a  flat 

sheet  of  it,  and  then  stamped 
into  the  required  shape  by 
means  of  a  machine  called  a 
"  die."  In  computing  the  size 
of  the  blank  to  be  cut  for  a  lid  of  given  size,  men  assume  that 
the  area  of  the  blank  is  equal  to  the  total  area  of  the  lid. 

Find  the  radius  of  the  blank  necessary  to  make  a  lid  whose  area 
is  12.5664  sq.  in. 

Suggestion.  —  If  i?  is  the  radius,  3.1416  R^  =  12.5664. 

19.  The  area  of  the  surface  of  the  lid  of  a  lard  bucket  is 
28.2744  square  inches.  Find  the  radius  of  the  blank  from  which 
it  is  made. 

74.    Product  of  the  Sum  and  Difference  of  Two  Terms.  — Find,  as 

in  Chapter  V,  the   products :     (a  —  4)  (a  +  4)  ;    (P  +  5)(P  —  5)  ; 
(a^  -  7)(x'  +  7) ;  (2E^  +  3)(2  R^  -  3). 

How  many  terms  in  each  product  ?  Why  not  more  ?  Why  is 
the  last  term  negative  in  each  product  ? 


SPECIAL  PRODUCTS  AND   QUOTIENTS 


128 


In  general,  where  a  and  h  are  any  two  terms,  it  is  found  that : 

(a  -h  b){a  -b)  =  d'-  b\ 
Hence, 

To  find  the  x)roduct  of  the  sum  and  the  difference  of  the  same  two 
terms,  take  the  difference  between  their  squares. 

Example.  —  {'iw^  +  S)(iw^-3)  =  (4w^)^-  32 

=  16w;6-9. 


EXERCISES 
Give  at  sight  the  products  of  : 

1.  n-{-3  and  ri  —  3.  18. 

2.  A  +  6  and  A -6.  19. 

3.  -v  +  2  and  v  —  2.  20. 

4.  a;  4-  5  and  a;  —  5.  21. 

5.  fe  —  4  and  fc  +  4.  22. 

6.  w  —  7  and  n-\-7.  23. 

7.  w  -{-  V  and  i«  —  v.  24. 

8.  2  a  +  3  and  2  a  -  3.  25. 

9.  5P  +  1  and5P-l.  26. 

10.  4:8-7  and  AS +  7.  27. 

11.  10-3i?andl0+3ie.  28. 

12.  1-a^  and  1  +  ar^.  29. 

13.  6  +  2vand  6  —  2v.  30. 

14.  2a;4-5  2;and  2  a;  — 5z.  31. 

15.  9M-\-3Tsind9M-3T.  32. 

16.  5a  — 4  6  and  5a +  4  6.  33. 

17.  (6  W-{-7  V){6  W-7  V).  34. 


12^-5a)(125^  +  5a).   • 

a2  _  9)(a2  +  9). 

aJi_4)(a:3  +  4). 

2^2_i)(2.42  4-l). 
7s2_4i2^(7s2^4^2)^ 

l-6a;^)(l  +  6a;*). 
a8_6»)(a»  +  6«). 

4  -  A)(A  +  4). 
l-6a;)(6a;  +  l). 
2M-{-7N)(-7N+2M). 
12-{-5t^)(-12  +  5t^. 
-10Z>3+9)(10i)«  +  9). 
a  +  l)(a-l)(a2  +  l). 
2-Tr)(2+  Tr)(4+  TT^). 
I_a;)(l+a;)(l4-a0(l+a;0. 


Give  at  sight  the  quotients  of  the  following  : 

35.  (a''-h'^^(a-\-b).  37.    (Z)^  -  1) -?- (Z)  -  1). 

36.  (x'-y^^(x-y).  38.    (16  -  P^)  ^  (4  _  ij). 


124  ELEMENTARY  ALGEBRA 

39.    {^h^-2^f)-^(;6h  +  ^i).  42.    (144?i«- l)--(12n-^  -  1). 

41.    (2/^_36)-(/-f6).  44.    (86  0^2-49  2/0-^  (6  a; -7  7/-). 

Solve  without  the  aid  of  pencil : 

45.  ^2  +  n  -  6  =  (n  -  2){n  +  2). 

46.  a;2  _|_  2aj  4-  3  =  (a;  +  3)(a;  -  3). 

47.  4:A^-  A -^1  =  (2 A +  7)(2 A- 7). 

48.  9  i?  (72  +  2)  =  (3  i?  +  6)  (3  i2  -  6). 

75.    Products  of  Arithmetical  Numbers  by  the  Rule  in   §  74.  — 

The  product  of  two  arithmetical  numbers,  of  which  one  is  less 
than  a  multiple  of  10  and  the  other  exceeds  this  multiple  by  the 
same  amount,  may  be  giyen  at  sight  by  the  rule  in  §  74. 
Thus,  58  X  62  =  (60  -  2)  (00  +  2)  =  60'^  -  2^  =  3600  -  4  =  3596. 

EXERCISES 

Find  mentally  the  products  of  the  following : 

1.  19x21.  7.   48x52.  13.  85x75. 

2.  28x32.  8.    53x47.  14.  88x92. 

3.  22  X  18.  9.    69  x  71.  15.  99  x  101. 

4.  39  X  41.  10.   72  x  68.  16.  97  x  103. 

5.  43  x  37.  11.   63  X  57.  17.  109  x  111. 

6.  34x26.  12.    77x83.  18.  118x122. 

19.  Find  the  cost  of  22  boxes  of  berries  at  18  cents  a  box. 

20.  Find  the  cost  of  17  dozen  eggs  at  23  cents  a  dozen. 

21.  Find  the  cost  of  28  pounds  of  butter  at  32  cents  a  pound. 

22.  Buckwheat  weighs  52  pounds  a  bushel.     Find  the  weight 
:)f  48  bushels. 

23.  Onions  weigh  57  pounds  a  bushel.     Find  the  weight  of 
63  bushels. 

24.  A  field  is  77  rods  wide  and  83  rods  long.     Find  its  area. 


SPECIAL  PRODUCTS  AND   QUOTIENTS  125 

76.    Product  of  Two  Binomials  with  One  Common  Term. 


x  +  1 

x  —  7 

x^7 

x-7 

a;  +  5 

x-5 

x-5 

x  +  5 

ar^H-   Ix 

x"-   7x 

a^-\-7x 

x'-7x 

+    5a;  +  35 

-   5x-\-S5 

-5a;-35 

-\-5x-S5 

x'  +  12x  +  '6b 

x'-12x-hS5 

x'-]-2x-35 

ic2-2x-35 

Tell  how  the  first  term  of  each  of  the  above  products  is  obtained. 
How  is  the  coefficient  in  the  second  term  of  each  obtained  ?  How 
is  the  third  term  of  each  obtained  ? 

In  general, 

{x  +  a){x  -j-  b)  =  X-  -{-(a-{-b)x  +  ab. 

That  is, 

To  find  the  product  of  two  binomials  having  a  common  term,  take 
the  square  of  the  common  term,  2)las  the  algebraic  sum  of  the  unlike 
terms  times  the  common  term,  plus  the  algebraic  product  of  the  un- 
like terms. 

Example  1.  — (n  +  2)(n  +  3)=  ?i2 +(2  +  3)n  +  2.3 

=  n2  +  6  w  +  6. 

Example  2.  —  («  -  7)(«  +  4)  =  «2  _|-(_  7  +  4)  <  +(-  7  •  4) 

=  <2  _  3  ^  _  28. 

EXERCISES 
Give  at  sight  the  products  of: 

1.  (a +  4)  (a +  5).       10.  (?7i  +  T)  (m  +  8).  19.  (d-l^(d-2). 

2.  {x  +  l)(x  +  3).       11.  (/ir+l)(^+10).  20.  (S-6)(S-12). 

3.  (M-\-2)(M-\-6).    12.  (a; +  8) (a; +  6).  21.  (v-4:)(v-10). 

4.  (2/ 4. 7)  (2/ +  4).        13.  (a -5)  (a -3).  22.   (d-16)(d-10). 

5.  (n  +  6)(n  +  l).       14.  (W-S)(W-2).  23.   (a +  5) (a -3). 

6.  (v-\-S){v  +  S).       15.  {R-6)(R-1).  24.  (a;+ 12)(a;- 8). 

7.  (t^4.)(t-9).         16.  (;-3)(^-12).  25.  (n  +  8)(n-4). 

8.  (A-\-7){A-{-3).     17.  (B-W){B-3).  26.  (F-3)(F+9) 

9.  (Z)  +  5)(Z>  +  5).     18.  {N-9)(N-7).  27.  {t-5)(t  +  S). 


126  ELEMENTARY  ALGEBRA 

28.  (6 -12)  (6 +  7).      34.  (m-|-16)(m-12).  40.  (y'i.7)(y'-ll). 

29.  (x  +  3)(x-15).      35.  (i6-ll)(?^  +  9).  41.  (c2-10)(c2  4-4). 

30.  (2/_l)(2/-f6).        36.  (C4-18)((7-7).  42.  (2a  +  5)(2a+3). 

31.  (^H-4)(JT-9).     37.  (a;2  +  2)(a;2  +  5).  43.  (3a;+7)(3aj-f2). 

32.  (i2-15)(i2+10).  38.  (a3-l)(a3  +  9).  44.  (6^-1)  (6  ^-4). 

33.  (^  +  14) (^-4).   39.  {P'-6)(P'-8).  45.  (8 x+3)(8a;-2). 

46.  (4  TT- 2) (4  Tr+ 5).  55.  (6k -{-3t)(6k-5t). 

47.  (5y  +  7)(5y  — 2).  56.  (4m  +  9n)  (4  m- 3  w). 

48.  (2c-3)(2c-12).  57.  (9  2) -3'y)  (9Z)  + 8 v). 

49.  (8'y  +  16)(8v-9).  58.  (7  v -{-2w)(Tv-9w). 

50.  {2x'  +  5)(2x'  +  S).  59.  (3 a^  +  4 6^)  (3  a^  + 12 fc^). 

51.  (7 ^3 -2) (7 ^3 +  6).  60.  (5t'-6s^)(5f-\-4.s^). 

52.  (12  TT^ - 20)  (12  W  +  16).      61.  (SB' -4.  AC) (8  B'  +  ^0). 

53.  (2a  +  36)(2a+4&).  62.  (4 c^ - 22 c?-') (4 c^  +  5 d^). 

54.  (5x-2y)(5x-6y).  63.  (1  -  5  a)(l  +  3  a). 

Find  mentally  the  indicated  quotients : 

64.  (a^^_5a;_^6)--(aj  +  2).  69.  (s^  +  4  s  -  60) -- (s  + 10). 

65.  (n'-An-\-3)^(n-l).  70.  ( TT^  +  TF- 72) - ( TT- 8), 

66.  (^2-2^-15)-(^  +  3).       71.  (2/2 +  9 2/ +  20) -(2/ +  4). 

67.  (v2_7'y  +  12)-^(v-4).  72.  (r^- 16r +  63)--(r  -  9). 

68.  (r2_y_;L2)--(r-4).  73.  (m2-3m-54)-(m  +  6). 

Solve  mentally : 

74.  (a;  +  3)(a;  +  4)=a:2  +  33. 

75.  (v  +  5)(v-3)  =  ('^  +  2)(?;-l> 

76.  (27i  +  7)(2n-6)=4(w2  +  2). 


SPECIAL   PRODUCTS  AND   QUOTIENTS  127 

77.  Products  of  Arithmetical  Numbers  by  the  Rule  of  §  76.  —  The 
products  of  many  arithmetical  numbers  may  be  obtained,  without 
the  aid  of  pencil,  by  the  rule  of  §  76. 

Thus,  87  X  92  =(90  -  3)(90  +  2)=  902  -  1  x  90  -  6 

=  8100  -  90  -  6 
=  8004. 

EXERCISES 
Find  mentally  the  following  indicated  products : 

1.   32  X  33.  6.   43  x  38.  11.   89  x  96. 

*  2.    51x52.  7.48x53.  12.87x91. 

3.  42x43.  8.   76x74.  13.    112x113. 

4.  64  X  &^.  9.   87  X  83.  14.   198  x  199. 

5.  39x42.  10.   98x92.  15.   251x253. 

16.  Find  the  cost  of  34  dozen  eggs  at  32  cents  a  dozen. 

17.  If  a  train  goes  42  miles  an  hour,  how  far  will  it  go  in 
48  hours  ? 

18.  Find  the  cost  of  34  yards  of  cloth  at  36  cents  a  yard. 

19.  How  many  square  feet  in  a  lot  92  feet  wide  and  98  feet 
long? 

20.  Oats  weigh  32  pounds  to  the  bushel.  Find  the  weight  of 
35  bushels. 

78.  Square  of  a  Binomial.  —  What  operation  is  indicated  by 
(?i  +  6)^  ?     Find,  by  multiplication,  the  values  of : 

(a;  +  3)^  (^-5)^  {m-\-nf',  {m-n)\ 

What  two  terms  in  the  square  are  positive  in  each  of  the  above 
cases  ?     When  is  the  second  term  of  the  square  negative  ? 
It  is  seen  that  for  any  values  of  a  and  6, 

(a-f  6)2  =  a2  +  2a6  +  62  and  {a -  bf  =- a" - 2  ab -\' b\ 


128 


ELEMENTARY  ALGEBRA 


These  identities  give  the  following  rule  : 

To  square  a  binomial  take  the  square  of  the  first  term,  plus  {or 
minus)  tivo  times  the  product  of  the  terms,  i^lus  the  square  of  the 
seco7id  term. 

Example.  _  (2  a^  -  3  by  =  (2  a^y  -  2(2  ^2)  (3  &)  4.  (3  5)2 
=  4  a*  -  12  a%  +  9  b^. 


EXERCISES 

Find  the  squares  of  the  following : 
12.    7w  —  3v. 


1.  n  +  3. 

2.  x-\-5. 

3.  P  +  2. 

4.  a -4. 

5.  W-6, 

6.  t-3. 

7.  m  4-  n. 

8.  x-y. 

9.  2  a +3  6. 

10.  5  r  —  4  s. 

11.  3(7+8i). 


2 

xy 

xy 

X* 

13.  5J5  +  9  y. 

14.  10  6 -c. 

15.  9F-^Sa 

16.  4/f-7A 

17.  12x-\-y. 

18.  ^-15R 

19.  11  TF+3a. 

20.  4:E  —  9n. 

21.  5  ^  +  16  if. 

22.  a;2  +  l. 

23.  tt2-3. 

24.  n-  +  5. 

25.  i2_10. 


26.  a^-2/l 

27.  A'-{-2B^ 

28.  3^2_4  32_ 

29.  5w;-^  +  2?;3. 

30.  y'  +  l. 

31.  4F-^. 

32.  z^-i-5. 

33.  12-7a2. 

34.  9  +  15i23. 

35.  2p^-q\ 

36.  1-8(^1 

37.  1^-^AC, 

38.  a^/  +  5. 

39.  A' -4:  A. 


-^-X- 


40.    Show  geometrically,  by  the  accom- 
panying diagram,  that 


SPECIAL  PRODUCTS  AND   QUOTIENTS  129 

41.  Show  geometrically,  by  this  diagram,  that 

Solve : 

42.  (^  +  4)2  =  ^(«4-3)  +  36. 

43.  (2  F+  3)2  =  (2  F+ 1)(2  F4-  3). 

44.  (4A;-7)2  =  (4A;-5)(4A;  +  3). 

45.  (?t  +  2)2-w2  =  ^_5. 


h X-M >f-WH 

K X H 


79.  Squares  of  Arithmetical  Numbers.  —  By  the  method  of  §  78, 
the  squares  of  many  arithmetical  numbers  are  easily  found  with- 
out the  aid  of  pencil. 

Thus,  622  =  (60  +  2)2  =  3600  +  240  +  4  =  3844, 

and  792  =  (80  -  1)2  =  6400  -  160  +  1  =  6241. 

EXERCISES 
Find  mentally  the  squares  of : 

1.  41.  5.   68.  9.   92. 

2.  59.  6.   89.  10.   99. 

3.  72.  7.   78.  11.   98. 

4.  38.  8.    83.  12.    101. 

80.  Square  of  a  Polynomial.  —  By  actual  multiplication,  it  is 
found  that : 

{a  +  b  +  cy  =  a"  -\-  b'' -\- c-  +  2  ab  +2  ac  +  2  be. 

In  general, 

Tlie  square  of  any  polynomial  equals  the  sum  of  the  squares  of  all 
of  its  terms,  plus  ttvo  times  the  algebraic  product  of  eoLch  term  into  all 
of  the  terms  following  it. 

Example.  —  (2  n2  -  3  n  +  5)2  =  (2  n^y-  +  (  -  3  n)2  +  (5)2  +  2  (2  «2)  (  -  3  «) 

-f  2(2n2)(5)  +  2(-3w)(5) 
=  4  n*  _|_  0  ,j-2  ^  25  -  12  7i3  +  20  n^  -  30  » 
=  4  n^  -  12  n3  +  29  n-  -  30  n  +  25. 


13. 

109. 

17. 

249. 

14. 

199. 

18. 

301. 

15. 

148. 

19. 

502. 

16. 

498. 

20. 

999. 

130  ELEMENTARY  ALGEBRA 

EXERCISES 
Write  down  the  squares  of  the  following  polynomials : 

1.  a—h-\-c.  5.   2w  —  6'y -I- w.      9.    a  +  &  +  c -f  d 

2.  2a;  +  3?/+2;.       6.   A^  +  2A-\-l.     10.    x-y-\-z-w. 

3.  m  —  2n—p.       7.   a;"  — 2a;  +  3.       11.   a^  — a^  — a  +  1. 

4.  ^  +  3jB-a      8.   r2  +  s2_i2.         12.    l  +  2P-3P2-f 4i*. 

81.  Quotients  of  the  Form  a^  —  b^  Divided  by  a  —  6.  — By  actual 
division, 

(a3_  ^3)  ^(^a-b)  =  a'-\-ab  +  b\ 

That  is, 

If  the  difference  between  the  cubes  of  two  terms  is  divided  by  the 
difference  between  those  terms,  the  quotient  equals  the  square  of  the 
first  term)  plus  the  product  of  the  two  terms,  plus  the  square  of  the 
second  term. 

Example.— ^^^~^^^^  =  (2  ay  +  (2  a)  (3  6)  +  (3  6)2 
2a  —  3  6 

=  4  a2  +  6  a&  +  9  62. 

EXERCISES 

Find  mentally  the  values  of  the  indicated  quotients : 

^    Z^-8  „     27^-1 

d-4.s  ' 

8  w^  -  27  v^ 
2w-Zv 

hM-N  ' 
64.  A^- 125  & 


1. 

a^-f 

x-y 

2. 

m^  —  n^ 

m  —  n 

3. 

u^-v" 

W  —  V 

4. 

a?-l 

x-1' 

K 

l_p3 

D- 

-2* 

^- 

27 

s  — 

3* 

&- 

-64 

B- 

-4  • 

125 

-f 

5- 

-y 

a«- 

-86« 

7. ^.  12. 


13. 


9.    ^^^ ^.  14. 


1-P'  ^^'     a-2b'  ^^'       4.A-5B 


SPECIAL  PRODUCTS  AND   QUOTIENTS  13] 

,^     1000r«-l  _^     8  -  a%^  „.     8a3-276« 

Jo.     — -— :: — .  ZK).     -— — .  A'k. 


lOr-1  2-ah  2a-^h^ 


2- 

-a6 

a«- 

-1 

a^- 

-1* 

m« 

-n« 

m' 

-n^" 

1- 

-^» 

19     '^y'-l  23     i:=-L'  27     ^-^-y" 


Find  mentally  the  following  indicated  products : 

28.  (a  -  6)  (a2  +  a^  +  &")• 

29.  (m  —  n)  (m^  +  mn  -f  tj^. 

30.  {W-V){W^+WV+V^, 

31.  (a:  -  1)  (»2  ^  a;  ^  1)_ 

32.  (2a-l)(4a2  +  2a+l). 

33.  (3a;-2y)(9ic2^62.2/  +  4y«). 

34.  (4s-50(16s'  +  20s«4-25^^. 

35.  (a^-l)(a;*4-ar^  +  l). 

36.  (1-2Z)2)(1  +  2Z)2  +  4Z>*). 

82.  Quotients  of  the  Form  a^  +  b^  Divided  by  a  +  6.  —  By  actual 
division, 

(flS  ^  ^3^)   _j.  (a  ^  ^)  ^  ^2  _  ^^  ^  ^2^ 

That  is, 

If  the  sum  of  the  cubes  of  two  terms  is  divided  by  the  sum  of  these 
termSy  the  quotient  equals  the  square  of  the  first  term,  minus  tlie 
(troduct  of  the  two  terms,  plus  the  square  of  the  second  term. 

Example. -l^l^^i^'  =  (5  Sf  -  (5  S)  (2t)  +  (20« 
=  2b  S^- 10  St +  4:  t\ 


132  ELEMENTARY  ALGEBRA 

EXERCISES 

Find  mentally  the  values  of  the  indicated  quotients  •. 
1_   ^  +  ^  ^Q    i^'  +  21v^  ^^    x'-^l 


m  +  71  2  m  +  ?i  W^  +  B 

4    «'  +  !-  13    27i>^  +  ^,  22    ^/±2Il'. 

a  +  1  *     3i>4-«  '    22/^  +  3^2 


5    •^  +  ^  14  e4^  +  125:y«  12oP«4-l 

2/  +  2'                     •       40.^  +  52/     '  .      *     5P2+1 

«    27  +  ^  -.^  1000 /r^  +  a^  „^  216 +  a¥ 

D.     — •  15.  — -•  >54. 


3  +  ^  lO/f  +  a  i5-{-at 


64  +  J\r3  216  771^4- 27  71^       ^^    343  + 8  v¥ 

^  +  N  6m  +  3w  7  +  2  vH^ 

g    0^^  +  125  6«  +  343d^  g^  +  JT^ 

i»  +  5    ■  •      h  +  ld    '  '  H^  +  K^' 

^    1  +  8^^  ^g    8P^  +  125c«  27 +  n^ 

1  +  26  •      2P  +  5c    *  '     3  +  w*  ' 

Find  mentally  the  following  indicated  products : 

28.  (m  +  n)(m2-mn  +  7i2).        33.    (4/S'  +  l)(16/S2- 4>S'  +  1). 

29.  (x  +  l){x'-x-]-l).  34.    (a  +  56)(a2-5a6  +  2562) 

30.  (l+a)(l-a  +  a2).  35.    {l  +  o^(l-x' +  x'). 

31.  (Tr+FXT^'-TFF+F^.   36.    {f^z^){y'-yh^  +  z''): 

32.  (3P+l)(9P2_,3p_|_i),   37.    (2+3^)(4-6^  +  9^2)_ 


SPECIAL  PRODUCTS  AND   QUOTIENTS  133 

SUPPLEMENTARY  EXERCISES 
Give  the  squares  of: 

2.   3  X« ;  5  X«+i  F«-' ;  - 12  a  =^6^ ;  -  7  V^-^  TF^'^+i 
Give  the  square  roots  of : 

4.    16//«+8;  25X8'*-2r«"+-;  81  a^"^-"+*. 
Give  the  cubes  of : 

6.  2^";  SiC+y-';   -4Jf*''iY2-;   -  5  c^-^d^"+*. 

Give  the  cube  roots  of : 

7.  w^";  Q*;  ^V^";  -^4«^;  -Sa^^Y". 

8.  27a^<»«;  -64^V";  -125a«"P^ 

Give  at  sight  the  products  of : 

9.  (a;'» -f  1)  (a;"  -  1).  13.    (^^  +  3)  (^'*  +  7) 

10.  (F'  +  3)(F*-3).  14.    (Z^  +  G)(Z>2'-4). 

11.  (a"+i  +  2)(a«+^-2).  15.    (xV- - 1)  (iV^*  +  8). 

12.  (a;- 4- 2/*)  (a;- -2/*).  16.    (3 /«  +  6)  (3 /~  -  2). 

Give  the  squares  of : 

17.  a'*  4-1.  19.    F'+T".  21.    5H^-d^. 

18.  af  —  y\  20.    2iy3«  +  3w;2«.  22.    3i92'^4-g2^ 

The  products  of  polynomials  often  may  be  obtained  mentally 
by  grouping  their  terms  so  as  to  form  binomials. 

Find  the  products  of  the  following  by  writing  the  factors  as 
binomials,  and  using  the  rule  in  §  74. 


134  ELEMENTARY  ALGEBRA 

23.  (a  +  h  +  c)(a  +  h  —  G). 

Solution. — (a  +  &  +  c)  (a  +  6  —  c)  =  ([a  +  &]  +  c)  ([a  +  6]  —  c) 

=  a2  +  2  a6  +  62  _  c2. 

24.  {x-\-y  —  l)(x-\-y-]-l).  27.    (6- c  +  6)(6- c- 6). 

25.  (a2  +  a  +  l)(a'  +  a-l).  28.   (>S'+2  ^-3)(^  +  2^  +  3). 

26.  (m  —  n-{-p)(m  —  n—p).  29.    (a  — 56  — c)(a  — 5  6  +  c). 

30.  (iV+3'y-5w)(JV4-3v  +  5ty). 

31.  (a-2  6  +  c)(a  +  26  +  c). 
Suggestion. —  Group  a  and  c. 

32.  (P-Q  +  i2)(P+Q  +  i2). 

33.  («2  +  a;  +  l)(a;2-a;  +  l). 

34.  (a^  -  a6  +  62)  (a^  +  «&  +  &")• 

35.  (a  +  6-c)(a-6  +  c). 

Solution.  —  Since  h  and  c  both  have  different  signs  in  the  two  factors, 
group  them.     Then 

(a  +  &-c)(a-  6  +  c)  =  (a +[6-  c])(a-[6-c]) 
=  a2-[&-cP 
=  a2  _  [62  _  2  6c  +  c2] 
=  a2  -  62  _)_  2  &c  -  c2. 

36.  (4n-3a;-y)(47i  +  3a;4-2/). 

37.  (2u  +  3v  —  w)(2u-3v  +  w). 

38.  (3P-4T+F)(3P4-4:r-F). 

39.  {a-\-h  +  c-\-d)(a-\-h—c  —  d). 

Suggestion,  —  Group  a  and  h  into  one  term,  and  c  and  d  into  one  term. 
Thus,  ([a  +  6]  +  [c  +  d])([a  +  6]  -  [c  +  tf]). 

40.  (x-2y  —  t-\-z){x-2y  +  t-z). 

41.  (P+2^-2!r+C^)(i2  +  2/S'  +  2r-V). 

42.  (ar'-a:2_a._|_i)(a^_ic2  +  a;_l). 

43.  (lH-a-a2-a«)(l  +  a  +  a2  +  a3). 


SPECIAL  PRODUCTS  AND   QUOTIENTS  135 

Find  the  products  of  the  following  by  writing  the  factors  as  bi- 
nomials and  proceeding  as  in  §  76 : 

44.  (a  +  &  +  3)(a  +  5-5). 

Solution.  —  (a  +  6  +3)  (a  +  6  -  5)  =  ([a  +  6]  +  3) ([a  +  6]  -  5) 

=  [a  +  &]-^  -  2[a  +  6]  -  15 
=  a2  +  2  a6  +  62  -  2  a-  2  &  -  16. 

45.  (aj-y  +  2)(a:-2/  +  3). 

46.  {p  +  q-\-lQ)){p  +  q-lQ). 

47.  (2  >F-  3  F+  8)(2  TF- 3  F-  3). 

48.  (aj2  +  a;4-6)(ic2^a._2). 

49.  (f-t  +  10)(f-t-U), 

50.  (n2-l4-n)(w^-2-|-w). 


CHAPTER  VIII 

FACTORS.     MULTIPLES.    EQUATIONS    SOLVED   BY 
FACTORING 

83.  Factoring.  —What  are  the  factors  of  6  ?  Of  10?  Of  12  ? 
Of  mn?     Oiv{v-\-t)? 

The  process  of  finding  the  factors  of  a  given  expression  is  called 
factoring. 

If  a  factor  of  a  given  expression  does  not  itself  have  factors,  it 
is  called  a  prime  factor  of  the  expression.  In  general,  to  factor 
an  expression  means  to  find  its  prime  factors. 

Facility  in  the  factoring  of  many  expressions  depends  upon  an 
ability  to  recognize  the  special  forms  of  the  expressions  and  upon 
a  knowledge  of  the  rules  in  Chapter  VII. 

84.  Polynomial  with  a  Monomial  Factor.  —  If  each  term  of  a 
given  polynomial  is  divisible  by  the  same  monomial,  the  poly- 
nomial is  divisible  by  the  monomial,  and  hence  the  monomial  is 
a  factor. 

For  example,  in  18  a*  —  9  a^  +  27  a^^  each  term  is  divisible  by  9  a^. 

2  a2  _  «  +  3 
9a2)i8a4_9a3  +  27a2 
Hence,  since  the  dividend  always  equals  the  product  of  the  divisor  and 
quotient, 

18 a* -  9 a^  +  27  a2  ^ 9 a^(2  d^-a-\-^). 
That  is,  the  factors  of  18  a^  _  9  ^3  +  27  a^  are  9  a2  and  2  a2  _  «  +  3. 

Hence  the  rule  for  finding  the  factors  of  a  polynomial  with  a 
monomial  factor : 

Find,  by  inspection,  the  monomial  of  highest  power  that  will  divide 
each  term  of  the  polynomial.  Divide  the  polynomial  by  this  mono- 
mial. The  divisor  and  quotient  are  the  monomial  and  polynomial 
factors,  respectively,  of  the  given  polynomial. 

136 


FACTORS.    MULTIPLES.    EQUATIONS 


137 


The  division  should  always  be  performed  mentally^  and  the 

work  written  as  shown  in  the  following  example : 

Example.  —  Factor  2  wx^  —  4  ny'^  +  6  nz^. 

2  /ix2  -  4  ny2  +  6  W2;2  =  2  n(a;2  -  2  2/2  4-  3  5;2) . 


6.  .43-4^2 

7.  TrR'-hTrR". 

8.  oi^  +  5x\ 

9.  n%^  —  m\ 

10.  2TrR^-\-27rRH. 


11.  sH^  +  s'l^, 

12.  5?^  — lOw;^^. 

13.  ISar'-giK^. 

14.  c^n-\-2<^n\ 

15.  7P^  +  21P3. 


EXERCISES 

Factor : 

1.  2a-\-2b. 

2.  3  a;  — 6?/. 

3.  AB'-ieAC. 

4.  5?ii2  +  10nQ. 

5.  8c?ic2_3^2/2. 

16.  a^  — a;2  +  a;. 

17.  4F12  +  6F1F2  +  4F1F3 

18.  H^-6H'b-^2H*b\ 

19.  'y2^  +  2vi2^_^^^ 

20.  8n«m»4-6wV  +  4nm^ 

21.  3a^&-6a362^3a263. 

28.  4a;«-6a;y  +  8fl;3/-12a.V. 

29.  -10M'i-12M'-16M*+SM\ 

30.  56  ay -14  ay +28  ay -105  ay. 

31.  Show  by  this  diagram  that  Qc^-{-icy^x(x  +  y). 
^ ^  32.    Show  by  a  diagram  that 

7ia 


22.  2TrR^-\-27t)''-\-2irRr. 

23.  a^6V  +  a26V  +  a^6V. 

24.  2^  li^d^- 12  yj'd  + 4:2  w'^dK 

25.  Gy^  +  Sv^O-lG-y*. 


26. 
27. 


—  a^  —  sc^  —  x. 


X^ 


i»-  OrlD—*^ 


nb  =  n(a  —  b). 
33.    Many  arithmetical  computa- 
tions may  be  shortened  by  applying 
the  process  of  factoring  in  §  84.     Thus, 

4  X  7  -f  6  X  7  =  10  X  7,  or  70. 
Find  by  this  method  the  values  of  the  following :  17  X  45  +  13 
X45;   38x92  +  42x92;    64x575  +  85x575-24x575;   9x 
3.1416  +  16x3.1416;    2x16x8  +  2x24x8  +  16x24;   365  X 
784  -  15  x  784  +  50  x  784. 


138  ELEMENTARY  ALGEBRA 

34.  If,  in  two  numbers  of  two  digits  each,  the  tens'  digits  are 
equal  and  the  sum  of  the  ones'  digits  is  10,  the  numbers  may  be 
expressed  by  10  ^  +  w  and  10  ^  + 10  —  «^,  respectively,  in  which  t 
is  the  tens'  digit,  u  the  ones'  digit  of  one,  and  10  —  w  the  ones' 
digit  of  the  other. 

Show  that  their  product  is  100  f  +  100  ^  + 10  u  -  u\ 

Show  that  this  may  be  written  100  t{t  + 1)4-1^(10  —  u). 

Hence,  to  multiply  two  numbers  of  which  the  tens'  digits  are  equals 
and  the  sum  of  the  ones^  digits  is  10,  take  the  product  of  the  number 
of  tens  by  a  number  one  greater  for  the  hundreds  and  annex  the 
product  of  the  ones. 

Thus,  74  X  76  =  5624  ;  69  x  61  =  4209  ;  43  x  47  =  2021. 

35.  Give  at  sight  the  products  of  the  following:  23  x27;  16  x 
14;  26  X  24;  32  X  38;  46  X  44;  ^6  X  54;  68  X  62;  77  X  73;  88  x 
82;  93x97;  81x89;  84x86;  79x71;  75x75;  91x99;  94x96. 

36.  State  a  rule  for  squaring  any  number  ending  in  5  that  fol- 
lows as  a  special  case  of  the  rule  in  Exercise  34. 

37.  Give  at  sight  the  squares  of  the  following : 

25;  45;  35;  65;  85;  55;  95;  75;  105;  115. 

85.  Polynomials  Factored  by  Grouping  Terms.  —  Some  polyno- 
mials that  do  not  have  monomial  factors  may  be  factored  by  the 
method  of  §  84,  after  first  grouping  the  terms. 

Example:  1.  —Factor  a^  -\-  ah  -{■  ac -\-  be. 

Grouping  the  first  two  terms  and  factoring  them,  then  the  last  two,  gives 
a(a  +  h)  +  c(a-\-  b). 

Now  the  factor  a  +  6  is  common  to  the  two  terms.  Dividing  by  a  +  6 
gives  a  quotient  a  +  c. 

Hence  the  factors  are  a  +  b  and  a  ■}-  c. 

That  is,  a2  + a6 +<zc  + &c  =  (a  +  6)(a +  c). 

Example  2'  —  Factor  mt  +  nv  —  nt  —  mv. 

Grouping  the  first  and  fourth  terms,  and  the  second  and  third  terms, 
mt  +  nv  —  nt  —  mv  =  m(t  —  v)—  n{t  —  v) 
=  (t  —  v)(m  —  n). 
Notice  that  we  factor  out  —n  from  the  second  group,  rather  than  +  n,  so 
that  the  expressions  left  in  parentheses  will  be  the  same. 


FACTORS.    MULTIPLES.    EQUATIONS  139 

These  examples  illustrate  the  rule : 

Group  the  terms  of  the  polynomial  to  he  factored  so  that  a  mono- 
mial factor  may  be  divided  out  from  each  group,  and  that  the  expres- 
sions left  in  parentheses  are  the  same  in  all  of  the  groups.  Then 
divide  by  the  expression  in  parentheses,  writing  the  divisor  as  one 
factor  and  the  quotient  as  the  other. 

If  in  Example  2  we  should  group  the  second  and  fourth  terms, 
factoring  out  v,  and  group  the  first  and  third  terms,  factoring  out 
t,  we  would  get  the  factors  {v—t){ii—m).  It  is  evident  that  these 
factors  differ  from  those  in  the  solution  above  only  in  their  signs. 
So  in  other  types  of  problems  in  factoring,  two  sets  of  factors  may 
be  obtained,  differing  only  in  sign.  For  practical  purposes  only 
one  set  of  factors  is  ever  needed,  and  the  other  may  be  ignored. 

Note.  — The  student  should  check  his  work  in  every  problem  in  factoring 
by  some  method.  In  most  cases,  this  is  easily  accomplished  by  multiplying 
mentally  the  factors  obtained,  to  see  if  the  product  equals  the  given  expres- 
sion to  be  factored.  A  second  method,  convenient  in  many  cases,  consists 
of  assigning  special  values  to  all  letters  involved  and  comparing  the  values 
of  the  factors  with  the  value  of  the  given  expression.  Thus,  if  in  Example  1 
above  we  make  a  =  l,6  =  2,  c  =  3,  then  the  expression  to  be  factored  be- 
comes 12  and  the  factors  3  and  4,  respectively. 

EXERCISES 
Factor : 

1.  S(x-\-y)-{-n(x-{-y).  9.  B^CN-J^P^+iACiM^IT). 

2.  a{P-Q)-\-b(P-Q).  10.  x{w-v)-\-(v-w), 

3.  v(a-\-b)-t{a-\-b).  H-  ac -\- ad -\- be  +  bd. 

4.  2x(m-n)-Sy(m^n).  12.  nm -f wy -f  mx  +  ay. 

5.  5w(Ji-k)  +  u(Ii-k).  13.  vt  +  wt-^2v-\-2w. 

e.  a(x-y)-^b{y-x).  1^-  Wg -  WJi+gk-hk. 

Suggestion.  -This  may  be  writ-  ^^'  mp^np-mq-  nq. 

tena(x-y) -6(a;-y).  16.  as  —  bs  —  at -\- bt. 

7.  s(A-5)  +  t{5^A).  17.  vu-vR-[-uS-RS. 

8.  p(r  —  E)  —  q(E  —  r).  18.  mn -\- mt  —  ns  —  st. 


140  ELEMENTARY  ALGEBRA 

19.  x^J^xy-\-xz-\-yz.  25.    h^k^  -  m^Tc^  -  h^ -\- m\ 

20.  p'^-\-pr—pq-qr,  26.    W^x-V^x—W^y  +  V^y. 

21.  M^-MN-\-MP-NP.  27.   i^(7+ (^0+ 22^^+2  G^. 

22.  2v-\-v^  —  2u  —  vu,  28.  pg  — 3^  — 2g4-6. 

23.  Zrs-2sx-\-3ry-2xy,  29.   R^-R  —  Rr^r, 

24.  2«-^m-2m  +  4.  30.   2 a^  —  ca;^ _ ^2^ ^ 2 a;^^ 

86.  Trinomials  that  are  Perfect  Squares. — If  a  binomial  is 
squared,  as  in  §  78,  the  result  is  always  a  trinomial.  A  trinomial 
which  is  the  square  of  a  binomial  is  called  a  perfect  square. 

Since  any  binomial  must  be  either  of  the  form  a  +  6  or  a  —  ?>, 
and  since  (a  +  hf  =  a^  +  2ah  +  V  and  {a -by  =  a^-2ab  +  b^,  evi- 
dently any  trinomial  which  is  a  perfect  square  must  be  either  of 
the  form  a^  +  2  a6  +  6^  or  a^  —  2  a6  +  b^.  In  either  case,  the  term 
2  ab  is  twice  the  product  of  the  absolute  values  of  Vo^  and  V6^. 
Hence, 

A  trinomial  is  a  perfect  square  if  one  term  is  twice  the  product  of 
the  absolute  values  of  the  square  roots  of  the  other  two  terms. 

In  9  ic2  —  24  xy  +  16  y^,  what  are  the  square  roots  of  9  x^  and  16  y^  ?  What 
is  twice  the  product  of  their  absolute  values?  Is  the  trinomial  a  perfect 
square  ? 

Since  the  binomial  factors  of  a  trinomial  which  is  a  perfect 
square  are  equal,  to  factor  such  a  trinomial  is  equivalent  to  express' 
ing  it  as  the  square  of  a  binomial. 

Example  1.— Factor  25  m^  _  40  m  +  16. 

V25»»2  =  ±  5  m,  and  Vl6  =  ±  4. 

Since  twice  the  product  of  these  roots  must  give  —  40  m,  the  positive  root 
of  one  term  and  the  negative  root  of  the  other  must  be  chosen,  which  can  be 
done  in  two  ways. 

Hence,  25  m^  -  40  m  +  16  =  (5  m  -  4)2  or  (-  5  m  +  4)2. 
Check.  —  When  m  =  2,  the  trinomial  is  36  and  the  factors  6  and  6,  or  —  6 
and  —  6. 

Example  2.  —  Factor  64  v^  +  112vt  +  49 1^. 

V64t?2  =  ±  8  V,  and  V4972  =  ±  7  «. 


,      FACTORS.    MULTIPLES.    EQUATIONS  141 

Since  twice  the  product  of  these  roots  must  give  +  112v«,  they  must  be 
taken  with  like  signs,  which  can  be  done  in  two  ways. 

Hence,  64v2  + 112 v«  + 49 <2=  (8v+70^or  (-gtJ-TO^. 

It  is  seen  here,  as  in  §  85,  that  there  are  two  sets  of  factors  in 
each  problem.  In  practice  only  one  of  these  is  used,  viz.  the  set 
in  which  the  first  term  is  positive. 

To  find  a  binomial  of  which  a  given  trinomial  is  a  square,  take 
as  the  terms  the  square  roots  of  the  two  square  terms  of  the  trinomial 
with  such  signs  that  their  product  gives  the  sign  of  the  remaining 
term, 

EXERCISES 

Determine  which  of  the  following  are  perfect  squares : 

1.  a;'^-|-Ga;  +  9v  9.    16 ir'-12  m  +  9t\ 

2.  n2  +  16  +  8w.  10.   49x2^42/2_20. 

3.  1-4:A  +  4:AK  11.   24  TP-lOTP^  +  l. 

4.  25^2 ^12^  +  1.  12.   a^-2a'b'-b\ 

5.  a:^  +  ax  +  a\  13.   m^  + 18  ma -f  81  a*. 

6.  9^2_6v  +  4.  14.    62-12  6cH-36c2. 

7.  M-\-4:MN+4:lP.  15.   9 p^  +  SS pq -{- A9  qK 

8.  r'-10rs  +  25^.  16.   Z)*  +  l  +  2i>^ 

Make  perfect  square  trinomials  of  the  following  by  supplying 
the  missing  terms : 

17.  ar^^_?_|.2/2.  24.  3G-\-?-12x. 

18.  a'-2ab-^?.  25.  a--? +  100. 

19.  ?-\-2mn+n\  26.  ?-{-l-\-x\ 

20.  4^2^  ?  +  l.  27.  4P34-1  +  ?. 

21.  16-24^+?.  28.  r^^- 12  rir^  +  r. 

22.  25v^-?  +9w^.  29.  ?-32JV^+?. 

23.  Tr^+49  +  ?.  30.    ?  +  4262  +  ?. 


14^  ELEMENTARY  ALGEBRA 

Factor : 

31.  a^  +  6a;-f9.  46.  4:  wh -{- SQ  wv^  +  81  if. 

32.  i22-4i2  +  4.  47.  M^~50MN-\-e25]Sr''. 

33.  ^2  +  8^  +  16.  48.  9n^  +  S0r^r2-\-25r2\ 

34.  25  — lOa  +  al  49.  a^  —  2  a^y- -^  xy*. 

35.  l  +  12n4-36n2.  50.  a*  +  4  a^fe^  _^  4  a^^*^ 
.36.  49TF2-28TF+4.  51.  P2_6PQ3_^9Q6 

37.  ^2  +  2^  +  1.  52.  81m*  +  36m27i4-4w2. 

38.  81-18^  +  ^2^  53.  25W'-40W'g-i-16g^, 

39.  a;^  +  4ic2  +  4.  54.  afy^  +  2x^y-{-x. 

40.  16-24Z)2  +  9D*.  55.  c-c?^  +  2  cdmn  +  m V. 

41.  ax^-6axy  +  9ay^  56.  i^V - 8  i?r^  + 16 «l 
(First  remove  monomial  factor  a.)  57^  24  v^  +  36  i;^  +  4  v. 

42.  2n2  +  20nm4-50m2.  68.  25 2^  +  81 2*  +  90 2^. 

43.  tS^ ~  30 tSV+ 225  tV.  59.  64  ^j^  +  49  ^/ -  112  ^1^2. 

44.  16£r2  +  56ird  +  49(^.  60.  36^3  +  144^  +  144^2^ 

45.  2562-60  6c  +  36c2^  61.  -l-6a-9a2. 

87.  Trinomials  of  the  Form  x^-{-ax-\-b.  —  By  the  principle  in 
§  76j  the  product  of  two  binomials  having  a  common  term  x  is 
always  a  trinomial  in  the  general  form  x^+  ax-\-b. 

Thus,  (X  +  5)(a;  +  7)  =  x2  +  12 jc  +  36. 
Here  a  equals  12  and  b  equals  35. 

It  follows  that  a  trinomial  in  the  form  x'^-{-ax-\-b  may  he 
factored  into  two  binomials  with  a  common  term  x,  if  for  the  other 
terms  of  the  factors  two  numbers  can  be  found  whose  algebraic  sum 
is  the  coefficient  a  and  whose  algebraic  product  is  b. 

Example  1.  — Factor  x^—^x  +  15. 

For  the  second  terms  of  the  factors,  we  are  to  find  two  numbers  whose 
sum  is  —  8  and  whose  product  is  +  15.    These  are  evidently  —  3  and  —  5. 
Hence,  x^  -  8  a;  +  15  =  (x  -  3) (a;  -  5). 


FACTORS.    MULTIPLES.    EQUATIONS  143 

Check.  —  When  x  =  1,  the  trinomial  is  8,  and  the  factors  -  2  and  —  4» 
respectively  ;  or  mentally  multiplying,  (x  —  3)(x  —  5)  =  x^  —  8  x  +  15. 

Example  2.  —  Factor  m^  —  4  mn  —  45  n^. 

We  are  to  find  two  numbers  whose  algebraic  sum  is  —  4  n  and  product 
—  45  n^.    These  are  evidently  —  9  n  and  +  5  n. 
Hence,  m^  —  imn~-^bn^={m  —  Qn)(^m  +  6^  n). 


EXERCISES 

Factor : 

1.  t^j^5t+6.  20.  2>2^14|)  +  48.  39.  aj*  +  2aj  — 3. 

2.  a2^5a4-4.  21.  5^  _  5  ^  ^  g.  40.  i?^  4-372-10. 

3.  n^^Sn  +  12.  22.  iY2-5iV^+4.  41.  K^-^SK-4:. 

4.  P2  +  9P+14.  23.  w;2_8w;  +  12.  42.  s^-S5  +  2s. 

5.  a^  +  9a;  +  20.  24.  Q^-9Q-\-U.  43.  2/^  +  3y-18. 

6.  /S^  +  O/S  +  S.  25.  z2_924.20.  44.  3a-28  +  a2. 

7.  v^  +  lOv  +  ie.  26.  «2_io^-f-9.  45.  C«-12  +  4a. 

8.  a^-\-10a-\-9.  27.  i)--10Z)  +  16.  46.  Sm  +  wi^-U. 

9.  i22  +  10R  +  24.  28.  F'- 10^+21.  47.  62.^3  6-40. 

10.  A:2  4_iofe  +  21.  29.  a^- 10 a +  24.  48.  Saj-SG  +  ar^. 

11.  F=^H-11F+18,  30.  T'-9T+S.  49.  ^-35-18. 

12.  c«2  +  ll(«+24.  31.  2/2-17^4-30.  50.  /-3(/-40. 

13.  Jl[f24- 11  3f 4- 30.  32.  H'-UH+^5.  51.  m'-U-5m. 

14.  62  4_i2&4-20.  33.  TF-4TF4-3.  52.  T^-4.T-12. 

15.  /4-12y  +  27,  34.  m2-12m4-32.  53.  0^2^53^4.52/2. 

16.  ^2^12^4-32.  35.  c2  4- 35 -12  c.  54.  a^  +  7ab+10b\ 

17.  r^  4- 15  r  4- 54.  36.  ^4-55-16  5.  55.  w;2_5^^^6^2 

18.  m^-{-17m-{-70.  37.  <2_f_7_8^.  56.  r^  -  7  rs  4- 10  s^. 

19.  0^4- 13  c  4- 30.  38.  i^2^2i?^-15.  57.  i^-3pq-2SqK 


144  ELEMENTARY  ALGEBRA 

58.  C^-^BCD^Un^,  66.  w^v^-\-2w^v-99i^. 

69.  f  +  Byz-Uz^.  67.  fZV^d%-56d3. 

60.  a'b'-^  Sab -IS.  68.  2  s^2_56  ^^^230^ 

61.  mV  — 2mn-35.  69.  3  a^  -  90  a;  +  600. 

62.  H^-19ir--120.  70.  6  62c  4-126  60  + 660  c. 

63.  F^-F- 30.  71.  SI  t'  + 30  at' +  aH\ 

64.  a62-3a6-70a.  72.  4:  W7i' -  2S  wn  -  240  w. 
(First  remove  monomial  factor  a.)         73.  90  v^/  —  204  v  +  6  vyK 

65.  iC2/2 ^ a;^/ —  90 a;.  74.  6-5a  — a^. 

88.  Trinomials  of  the  Form  ax' +  bx+  c.  —  There  are  different 
methods  of  factoring  trinomials  of  the  form  aa;^  +  6a;  +  c.  Two 
methods  are  given  here. 

First  Method.  —  If  any  two  binomials  such  as  mx-\-p  and 
nx  +  q  are  multiplied,  their  product  equals  mnx^  -{-  inqx  +  npx 
■i-pq,  which  may  be  written  mna^  +  (mq  +  np)  x  -\-  pq.  This  is  of 
the  general  form  aa^  +  6a;  +  c. 

Thus,  (2 a;  +  3)(3a;  +  1)  =  6 x2  +  11  x  +  3. 

If  a  trinomial  of  the  general  form  aoi^  -\-bx-\-c  has  binomial  fac- 
tors of  the  forms  7nx  -\-p  and  nx  -f-  q,  these  may  be  found  by  trial. 

Example  1. —  Factor  2x^  +  5x  -\-2.  The  first  terms  of  the  binomial  fac- 
tors must  be  factors  of  2  x^j  and  the  second  terms  must  be  factors  of  2. 
The  sum  of  the  products  of  the  first  term  of  each  by  the  second  of  the  other, 
called  the  ci'oss  products^  must  be  6  x.  Since  all  signs  of  the  trinomial  are 
positive,  only  positive  terms  for  the  binomial  factors  need  be  sought. 
The  possible  trial  sets  of  factors  may  be  arranged  as  follows : 
2a;  +  2  a;  +  2 

x+l  2a;+  1 

It  is  seen  that  the  second  of  these  gives  the  right  sum  of  cross  products. 
Hence,  the  factors  are  rK  +  2  and  2x  +  \. 

Check.  — 'WhQTi  a;  =  1,  the  trinomial  is  9,  and  the  factors  3  and  3,  respec- 
tively ;  or  multiplying  the  factors,  (a:  +  2)  (2  a;  +  1)  =  2  x^  +  5  x  +  2. 

Example  2,  —  Factor  ^vfl  -{■  lov  —10  v^. 

Since  the  third  term  is  negative,  the  factors  of  it  chosen  must  have  un- 
like signs. 


FACTORS.    MULTIPLES.    EQUATIONS  145 

Four  of  the  many  possible  trial  sets  of  factors  are : 

W  —  2.V  io  +  2tj  w—^v  io+5o 

3to4-5t>  3to  —  5t)  810  +  2?)  3io  —  2d 

It  is  seen  that  the  second  of  these  gives  the  right  sum  of  cross  products. 
Hence,  Zvfi  +  wv  -  \Q  v'^  =  (w +  2v)(Z'w- bv). 

Each  trial  set  should  be  tested  as  written.  If  done,  it  usually 
will  be  unnecessary  to  write  down  all  possible  sets.  Each  set 
should  be  written  down  as  above,  and  the  sum  of  cross  products 
found  mentally.  .  In  a  short  time  the  student  should  be  able  to 
factor  most  trinomials  in  very  few  trials.  The  following  princi- 
ples systematize  the  work : 

I.  If  the  trinomial  contains  no  monomial  factor,  neither  factor 
can  contain  one. 

Thus,  in  Example  1,  since  2x^+  5x  -\-2  contains  no  monomial  factor,  it 
was  useless  to  write  the  first  trial  set,  with  the  binomial  2  x  +  2. 

II.  If  the  last  term  of  the  trinomial  is  +,  the  last  terms  of  the 
factors  tvill  be  both  +  or  both  — ,  according  as  the  middle  term  of 
the  trinomial  is  +  or  —. 

III.  If  the  last  term  of  the  trinomial  is  — ,  the  last  terms  of  the 
factors  will  have  unlike  signs. 

Second  Method.  —  In  mnx^  4-  '^V'Q^  4-  'npx  4-  PQf  which  is  the 
product  of  mx-\-p  and  nx  +  g,  the  product  of  the  terms  mnx^  and 
pq  equals  the  product  of  the  terras  7nqx  and  npx.  From  this  we 
may  infer  that 

If  we  may  separate  the  second  term  of  a  trinomial  in  the  form 
ax^  4-  6jr  4-  c  into  two  terms  ivhose  product  is  equal  to  that  of  the  first 
and  third  tenns,  we  may  factor  the  expression  as  in  §  85. 

Example  3.  —  Factor  3  x^  4  11  a-  +  6. 

3  x2 . 6  =  18  ic2.     Now  18  a;2  =  9  x .  2  X  =  18  sc .  x.    The  sum  of  the  factors 
9  X  and  2  x  is  the  middle  term.     Hence  we  write 
3x2  +  llx  +  6  =  3x2  +  9x  +  2x46 

=  3x(x  +  3)  4  2(x4  3) 
=  (x  +  3)(3x4  2). 
Note  that  it  will  be  necessary  only  to  use  the  coefficients  in  determining 
how  to  break  up  the  middle  term,  as  shown  in  the  next  example. 


146  ELEMENTARY  ALGEBRA 

Example  4.  —Factor  12  «2  _  5  «?,  _  2  v^. 

12(-  2)  =  -  24.     24  =  3  X  8  =  6  X  4  =  etc.     Since  the  product  -  24  has 
a  negative  sign,  the  two  terms  into  which  —  btv  \^  to  be  separated  must 
have  unlike  signs.     The  required  terms  are  —^tv  and  3  tv. 
Hence,       12  f^ -  btv  -  2  v'^  =  12  t'^  -  ^tv  +  ^tv  -  ^v"^ 

=  4  «(3 « -  2  v)  +  v{?j  «-2 1^ 
=  (3«-2?j)(4«  +  v). 

EXERCISES 
Factor : 

1.  10a2  +  17a  +  7.  22.  452-35-27. 

2.  3^2^11^  +  10.  23.  6n2-17n-14. 

3.  3a;2  +  10a;  +  3.  24.  ^lJ-2L-n. 

4.  5m2  +  12m  +  4.  25.  3i^2-i^-4. 

5.  2F2_^9F+10.  26.  SR^-R-2. 

6.  3^2^10^  +  7.  27.  2a'^x-ax-2Sx. 

7.  8  7^  +  22  r  +  15.  (First  remove  the  monomial  fac- 


8.   3B^-11B  +  10, 


tor.) 


9.  10y"-17y-h7.  28.  6c^d-cd-S5d, 

10.  3S^^10S-\-S.  29.  6z''-7z-20. 

11.  5^-12^  +  4.  30.  20a;2-46a;-84. 

12.  15-22a;  +  8a^.  31.  S6g^-15g-6. 

13.  2P2-9P+10.  32.  4a2/2-10a^-36a. 

14.  5-llc  +  2c2.  33.  WW'g-5Wg-20g, 

15.  4-8A;  +  3fe2^  34.  SO  w^v^  -  37  wv^  -  7  v\ 

16.  4jR2  +  3i2-27.  35.  2QH-5Qt-25L 

17.  6m2  +  17m-14.  36.  4  02  +  32(7  +  15. 

18.  3i>2  +  2Z)-5.  37.  4a^  +  23a;2/  +  lS/- 

19.  3a2  +  a-4.  38.  2  M^  +  15  MN-['7  N\ 

20.  3^2  +  ^-2.  .39.  2d^-3dk-l-¥. 

21.  4ir2_^_5.  40.  6.£;2__23^F+20i<^2 


FACTORS.    MCLTIPLES.    EQUATIONS 


147 


41.  U  1^-^-2(^-11  be, 

42.  7  A^  +  4.n^-29An, 

43.  21i22  +  2i?^   -8^^ 

44.  2ri2_5n^'2-18rs* 

45.  ST^S-2  TT'S~.   r"S. 


46.  12  G'rv'- 5  Gdn'- 2  cPn\ 

47.  4:  b'^x- 2  bdx- 20  cPx. 

48.  12  r^-4rz-i6Z2. 

49.  3  a* -a^ -2. 

50.  6  2<;2^  +  15w;?;2f-54'y*f. 


89.  Binomials  of  the  Form  a^  —  b^.  Since  the  product  of  a  -f  6 
and  a  —  b  is  a-  —  b^,  thts  factors  of  a^  —  b^  are  a +  6  and  a  —  b. 
Hence, 

27ie  factors  of  the  difference  between  the  squares  of  two  numbers 
are  the  sum  and  the  difference  of  the  numbers. 
Example.  — Factor  4  <2  _  25. 

4  <2  _  25  =  (2  0*  -  (5)2 

=  (2«  +  5)(2<-5). 
Check.  —  When  <  =  1,  the  trinomial  is  —  21,  and  the  factors  7  and  —  3 
respectively.     Or,  check  by  finding  the  product  of  the  factors. 

Observe  that,  by  taking  the  negative  square  roots  of  the  terms,  we  might 
also  write  4 ««  _  25  =  (  -  2  0^  -  (-5)2 

=  (_2(-5)(-2«  +  5). 
See  §  84  and  §  85. 

EXERCISES 
1.   Show  by  the  accompanying  diagram  that  3^  —  y^=(x  +  y) 

(x-y)' 


Factor: 

2.   N'-A. 

8. 

fc2--25. 

3^ 

3.    a^-9. 

9. 
10. 
11. 
12. 

16 -S'. 
H'-64. 
49 -wl 

100 -?;2. 

X 

1. 

4.  F'-l. 

5.  ^2-16. 

1 

6.    1-P\ 

K X- 

— *— y — ^ 

7.   4-wK 

13. 

ar'-81. 

14.   A'-IU.^ 

16.    1- 

-49^^ 

18.    h'-l<^. 

16.    l-16a;2. 

17.    9- 

-42 

Jl 

19.    8] 

Lm2~64?i« 

148 


ELEMENTARY  ALGEBRA 


20.  a?  —  ^^y\ 

21.  16^2__25Z>2. 

22.  4  c^- 25(^2. 

28.  a^/-2^. 

29.  'm;^  —  w. 

30.  6at^-'24:a, 

31.  16m*-36m^ 

32.  HH-'2bf. 


23.  36jE;2„j.2, 

24.  81F2-<?9i*. 

25.  O^^lOODl 


26.  16a;2_i21/. 

27.  50  a2- 32  62. 
(Remove     the    mo- 
nomial factor  first.) 

S3.  28Fy-7^^ 

3<.    ,/Y_l. 

86.   4-36a2^2^ 

86.   200P2j^2_3j^4^ 

37.    m'^  —  w*. 


38.  16 -T*. 

39.  a^  — 1. 

40.  TTfi^  — TTT-g^. 

41.  2  7rE'-S^RHK 

42.  Show  that  the  area  of  the  shaded 
border  in  this  figure  is  S  =  (A  -{-  d)(A  —  a). 

43.  Show  that  the  area  of  the  ring  shown  in  the  figure  is 
S  =  7r(E  +  r)(E  —  r),  where  E  is  tlie  outer  radius  and  r  the  inner 
radius  of  the  ring.  A  ^  0(^  P" 

44.  The   cross   section   of    a  hollow    iron//^j 
column  is  10  inches  in  outer  diameter  and  8 
inches  in  inner  diameter.     Find  the  area  by  IS  I 
the  formula  in  Problem  43. 

45.  The  cost  of  steel  construction  such  as 
is   used  in   bridges   and   modern   large   city       xZ5 
buildings  is  computed  at  so  much  per  pound  of  steel  used,  and 
hence  the  weight  of  steel  is  always  computed. 

A  hollow  steel  column  is  16  ft.  long,  14  in.  in  outer  diameter, 
and  10  in.  in  inner  diameter.  Find  its  weight,  allowing  490  lb.  to 
a  cubic  foot.     Use  formula  in  Problem  43. 

46.  Find  mentally  the  values  of  the  following: 

282 -222  J  372  _  332.  49'2-412j  76^-742;  972-932;  1162-114* 


FACTORS.     MULTIPLES.     EQUATIONS  149 

90.    Polynomials  whicn  can  be  Written  in  the  Form  a^  —  6^.  — • 

Some  polynomials,  by  gr(  •  iping  terms,  can  be  written  in  the  form 
a?  —  6^,  and  hence  f actorec  by  the  method  of  §  89. 

Example  1.  — Factor  lo^  4  j  wv  4- 1>2  —  t\ 

Grouping  terms, 

vfi  +  2  wv  ■{•  v"^  -  '    - :  Ctp2  +  2  tc»  +  t?^)  -  ^2 

Check.  — When  «?  =  1,  r=:l,«  =  l,  the  polynomial  is  3,  and  the  factors 
3  and  1,  respectively. 

Example  2.  —Factor  A^  -  B^  +  2  SC  -  C'\ 

A^  -  B^  +  2  BC  -  C^  =  A^  -(B^  -  2  BC  +  C2) 
=  A^  -{B-  0)2 

=  {A-\-B-C){A-B+  C). 

Learn  to  omit  all  except  the  first  and  last  stepSy  and  to  write  out 
the  work  as  follows  : 

Example  3.  —  Factor x^ +  6x— m^  +  4mn  —  in^ +  9. 
a;2  +  6  a;  -  »w2  +  4  m7i  -  4  n2  +  9  =  (x2  -f  6  a;  +  9)  -  (m2  -  4  win  +  4  w'-^) 

=  (a;  +  3  +  m  -  2  n)(a;  +  3  —  w  +  2  n). 

EXERCISES 

Factor : 

1.  (a  4- 6)^-1.  10.  i9V^-(ST-S)\ 

2.  {x-yy-z\  11.  (x-hyy-(a  +  by. 

3.  (S-hTf- 25.  12.  (M-N)'-(W-Vy. 

4.  (/t_A:)2-l6«».  13.  (C+Dy-'(4:E'-'Fy. 

5.  (2m  +  ny-S6.  14.  (l-xy-(v^ty. 

6.  1-(A  +  By.  15.  m2  +  2mw  +  7i2-62. 

7.  4:-(2w  +  vy.  16.  R'-4:RA-^4:A^-B\ 

8.  25a^-(b-cy.  17.  l+2x  +  x^^yK 


9.    36?f-(r2-4r3/.  18.   r^^  ■}- 6  r^r^  +  9  r^^ -- r^- 


150  ELEMENTARY  ALGEBRA 

19.  x*4-4a^4-4a;2_l.  22.    i -^  a" -2ab -h\ 

20.  W^-{-9D^-4:M''-eWD.        23.   i  -  M' -\-6  MV-9  V^ 

21.  l-8a;  +  16a;2-;32^  24.    1  -  A^  +  14  M  -  49A;2. 

25.  A^  -9B''  -30  BC  ~2n  C^ 

26.  16  C/^  +  36  FTF-  4  V-  81  TT*. 

27.  S^-  r2  +  4ar-4a=. 

28.  12i)g  +  16r2-4i)2_9  52 

29.  A'-AC'-SAB  +  16B'. 

30.  6^m-«2_9^2^j^^ 

31.  a2_|_2a6  +  62_c2_2c(^-(2«. 

32.  0/"^  —  4  ic  4-  4  —  m''  +  6  ?n,w  —  9  nK 

33.  l  +  2a;2/  +  2jp-2/^-a^4-y. 

34.  TF2  +  12a^-10  TFc^-4a2  +  25/-9<l 

35.  M^ -  12  dM- 4:kN+ 36 d^- A  N^-Jc". 

36.  a;^-^2^10^m2-16a;27i_25m^  +  64n*. 

91.  Binomials  of  the  Form  a^  —  b^.  —  From  §  81  it  follows  that 
the  factors  of  a^  —  b^  are  a  —  b  and  a^ -\- ab  +  b\  Test  these 
factors  by  actual  multiplication.     Hence, 

One  factor  of  the  difference  between  the  cubes  of  two  terms  is  the 
difference  between  the  terms,  and  the  other  factor  is  }he  sum  of  their 
squares  and  their  product. 

Example.  —  Factor  8  m^  —  126  #. 

8  m3  -  125  w3  =  (2  mY  -  (5  n)8 

=  (2  m  -  5  n)  (4  m2  +  10  mn  +  25  n^). 

C^ecA;.  —  When  m  =  1  and  w  =  1,  the  binomial  is  —  117,  and  the  factors 
—  3  and  39,  respectively. 


FACTORS.    MULTIPLES.    EQUATIONS 


151 


1. 


EXERCISES 

Show  by  the  accompanying  diagram  that  a^  —  y^={x-y) 


(x^^xy  +  f).      (If   f 

is   taken 

from   ic^,   it    leaves   three    slabs 
each  X—  y  thick.) 

Factor :                                      y^ 
2.   m^-n\                             /      ,, 

4 

'f-;T 

A 
d 

1 

..- 

z^ 

3.    a^-8. 

^x-y^ 

r-l- 

--^-U 

-- 

X 

4.    A^-\. 

1 

1     1 

1                     a 

1 

1 
1 

5.  1-F\ 

6.  8-r«. 

7.  27-^. 

8.  ^vy^  —  t^. 

1 

1 
1 

1 

j  / 

/ 

/\ 

1 
L. 

/ 

/ 

" 

" 

► 

9.   T3-27  V^. 

/ 

^ 

10.   64?/3-l. 

H 

— X— 

11.   3/3_i25iV^ 

16.   27  a;*- 

-«. 

21. 

8^«-27. 

12.    8n«-27r/. 

17.   125  a^ 

-a\ 

22. 

1,6  _  64  m;^ 

13.    64sr'-27a». 

18.  2Pe- 

- 16  P^ 

23. 

125  -  Z^. 

14.   216F«-125Fi^ 

19.   81 2/*- 

■24  2/. 

24. 

27  i2«-F«. 

15.   a%^-l. 

20. 

w?  —  n 

6 

25. 

4,r7^ 

-327r9'A» 

26.   The  volume  of  a  sphere  whose   diameter  is  D  is  ^-rrJ^. 
Show  that  the  volume  of  a  spherical  s/ie/i  whose  outer  diameter 

is  D  and  inner  diameter  d,  is 
F=  I- 7r(i)- €0(1^+ i)(^4- fi"). 

27.  Compute,  by  the  formula 
in  Problem  26,  the  volumes  of 
spherical  shells  whose  respec- 
tive outer  and  inner  diameters 
are  in  the  following  table. 


D 

d 

4  in. 

6  in. 

12  in. 

15  in. 

3  in. 

4  in. 

10  in. 

12  in. 

152 


ELEMENTARY  ALGEBRA 


92.  Binomials  of  the  Form  a^  -t-  61  —  From  §  82  it  follows  that 
the  factors  of  a^  +  W  are  a  +  6  and  a-  —  ah-\-  b^.  Test  these  fac- 
tors by  actual  multiplication.     Hence, 

One  factor  of  the  sum  of  the  cubes  of  two  terms  is  the  sum  of  the 
terms,  and  the  other  factor  is  the  sum  of  their  squares  minus  their 
product. 

Example.  —  Factor  27  W^  +  6iV^. 
27  >r8  +  64  F8  =  (3  Wf  +  (4  vy 

=  (3  Tr+4F)(9Tr2-12  TFF+16F2). 
Check.  —  When  W=l  and  F=  1,  the  binomial  is  91  and  the  factors  7  and 
13,  respectively. 


^ 

V 


;:« 


^ 


^ 


Factor : 

1.  m*-fw'. 

2.  A'+l, 

3.  l-^p^ 

4.  w^+S. 

5.  ^3  +  64. 

6.  27  +  ^8. 

7.  125  +  ^. 

8.  i)^  +  216. 

9.  1  +  8  a\ 
10.    l+27a^. 


EXERCISES 

11.  64JV3_|.l. 

12.  8a3+63. 

13.  2/^+272^. 

14.  TF^  +  64  2)^. 

15.  27/^^  +  8  T^. 

16.  125ri3  +  27r/. 

17.  64a3+27^. 

18.  W  0^+54:  y\ 

19.  a^ft^  +  l. 

20.  27+mV. 


21.  27  P2+P^ 

22.  8x*+125x. 
23.-  81  Z)*-i-  24  D. 

24.  'y^  +  ^«. 

25.  8i\r6-|,l. 

26.  40  a'' +  135  63. 

27.  7rZ>3  +  Trd\ 

28.  54Pi»+2i?/. 

29.  STrr^  +  81  TrrhK 

30.  to^  +  t'^ 


93.  General  Suggestions  on  Factoring.  —  As  shown  in  the  preced- 
ing sections,  the  method  of  factoring  any  given  expression  depends 
upon  the  special  form  of  the  expression.  There  are  no  general 
rules  or  principles  that  can  be  laid  down  for  factoring  all  expres- 
sions. To  factor  any  expression,  wherever  the  need  for  it  is 
encountered,  requires  first  that  the  expression  be  identified  as  hav- 
ing some  special  form,  such  as  one  of  those  discussed  in  this  chap- 
ter, and  second  the  application  of  the  special  rule  for  factoring 
expressions  of  that  form.     The  following  miscellaneous  exercises 


FACTORS.    MULTIPLES.     EQUATIONS  153 

are  given  for  practice  in  identifying  tlie  special  forms  of  expres- 
sions to  be  factored.  • 

General  Suggestions : 

I.  In  factoring  any  expression,  first  see  if  there  is  a  monomial 
factor;  and  if  there  is  one,  separate  the  expression  into  this  mono- 
mial and  the  corresponding  polynomial  factor. 

II.  T7ien  try  to  identify  the  polynomial  factor  as  having  one  of 
the  special  forms  discussed  in  the  preceding  sections;  and  if  success- 
ful, factor  it  by  the  appropriate  rule. 

III.  Repeat  the  process  with  each  factor  obtained,  until  the 
prime  factors  of  the  given  expression  are  found.  Write  the  given 
expression  as  the  indicated  product  of  all  of  the  prime  factors, 

MISCELLANEOUS  EXERCISES 
Factor : 

1.  A'-hSA'-^A\  16.  3P-{-lSM-\-42. 

2.  2a^y-i-6a^f-{-4:xf.  17.  42-13i/-|-/. 

3.  4i^  +  12wjv  +  9v*.  18.  F^_42-TK 

4.  m^n-^6mn-\-9n.  19.  a^  —  A2-\-a. 

5.  P'*  +  8P+12.  20.  ^-4^2_^4«. 

6.  ^2^3^^  12.  21.  8-^. 

7.  22H4i2-12.  22.  iB»  +  2 ar^ - 48 a?. 

8.  m2_4^_i2.  23.  Q^-27?-i&x. 

9.  a3  +  8a2-fl5a.  g^.  W^-9Wg  +  20g\ 

10.  M^-^M^'  +  l^M.  Sfi.  TT^H-Q  %4-20^2. 

11.  d^'\-2d^-lbd.  26.  H*-Ut^. 

12.  v^  — -u^  '  27.  a3  +  a2-h8a  +  8. 

13.  ^a?y-10^f  +  ^xy^.  28.  1  -  A.  m^ -{- A:  mv  -  v^. 

14.  a  —  a\  29.  x^y -\-Zxy —  2^y. 

15.  D'-2D''-1^D.  30.  12  +  N^  +  n  N. 


154  ELEMENTARY  ALGEBRA 

31.  11  at-{-12t-\-aH.  59.    V^+V^-^2V, 

32.^2^5^-50.  QQ.    o?  +  2xy-d5y\ 

33.  A^-^A-60,  61.   a^-{a-Q>)\ 

34.  2i»2  +  3a;4-l.        '  62.  .^V-i>'- ^'  +  1. 

35.  ^-125.  63.    7w;2  +  36w;  +  5. 

36.  aV-6W.  64.   2  ^2^  5  ^-3. 

37.  U4.U-M\  65.    2yl2_5^_|.3, 

38.  ax  —  hx  —  a  +  h.  66.    a2  4-4  a6  —  0^+4  ft^. 

39.  'y2_i5^^50.  67.   ^  +  125. 

40.  a^y^-lOxy-be,,  68.    5m^-^mn-2n\ 

41.  2a2-3a  +  l.  69.   812/^-3. 

42.  Q^i«-9F^.  70.  12  5-2 +  5  5'- 2. 

43.  xy-x  —  y-\-l,  71.   216  +  ^. 

44.  m^  +  Ti^  72.   Saha?  +  2abx—ab. 

45.  2A^  +  bAB-{-2BK  73.   SD^  +  l. 

46.  -v^  +  S.  74.    6  6c-lls^  +  5^2^ 

47.  wV  +  5  w;v^  +  6  ^^  75.    81  x*  - 100  v*. 

48.  D^-liy'  +  lO,  76.   JV^2_f.^^_2ivr_2a. 

49.  a^y  +  7  ar/ 4- 10.  77.    4  a^- (a +  6)2. 

50.  F'*  +  llF'-26.  78.   27^3  +  1. 

51.  6a2  +  lla  +  3.  79.   4  ^^  _^  32  ^5  +  39  ^2. 

52.  12w'-2Sw  + 10.  80.   125  v^  +  ^^ 

53.  27 +r^  81.   4:(a-bf-(y  +  zy. 
64.  36  3f2-64iV^J/2  g2.   8^^  +  15^-2. 

55.  Sx'^+7xy  +  2y\  83.   ri^  +  G  rir2-l-4  r3-4rs2  +  9  r/. 

56.  SlJ^^-ieC*.  84.    40Jf3  +  135F«. 

57.  ab-3a-2b  +  6.  85.    H^-f  +  vH-vt. 

58.  2m2-9m72  +  10n2.  86.   3x^-\-x-^2, 


FACTORS.    MULTIPLES.    EQUATIONS  155 

87.  A^-itA^n-'A-n,  94.  l-\-4:aH'^ -a^ -^1*. 

88.  15  +  22-22.  95.  9a2  +  66d-d2_952^. 

89.  xy  ^z^  —  xz  —  yz.  96.  v^  —  216  f. 

90.  9x2 -27  a; +  20.  97  5  Q2_  79  Q^_  16  ^2, 

91.  i>3-216.  98.  oj  +  a^-ox^-a. 

92.  TT^— F^  99.  xP ■\- v^w -\- vw^  +  in^. 

93.  mn2  +  7?7in-30m.  100.  R"^ -\-2  R- Rr -'2r, 

HIGHEST  COMMON  FACTOR  AND  LOWEST  COMMON 
MULTIPLE 

94.  Highest  Common  Factor.  —  An  expression  that  is  a  factor 
of  each  of  two  or  more  expressions  is  called  a  common  factor  of 
those  expressions.  The  expression  with  the  greatest  numerical 
coefficient  and  the  highest  powers  of  its  literal  factors  that  is  a 
common  factor  of  two  or  more  expressions  is  called  their  highest 
common  factor  (H.C.F.). 

We  shall  consider  here  only  the  highest  common  factors  of 
expressions  whose  factors  may  be  obtained  by  the  methods  of  this 
chapter. 

Example  1 .  —  Find  the  H.  C.  F.  of  8  A^BG^  and  12  A^B^CK 

The  largest  number  contained  in  both  8  and  12  is  4. 

The  highest  power  of  A  contained  in  both  A^  and  A^  is  AK 

The  highest  power  of  B  contained  in  both  B  and  B^  is  B. 

The  highest  power  of  C  contained  in  both  C*  and  C^  is  C. 

Hence,  the  H.C.F.  is  iA^BC^. 

Example  2.  —  Find  the  H.  C.  F.  of  x^  -  x^y  -  xy^  +  y^  and  2x8-  ix^y-\-2xy^. 

x»  _  a;2y  _  xy^ -{■  y^  =  (x  -  yY  (x  +  y). 

2x8  -  4x2y  +  2  ary2  _  2  X  (x  -  yy. 
Hence,  H.  C.  F.  =  (x  -  y)2,  or  x^  -  2  xy  +  y^. 

To  find  the  highest  common  factor  of  two  or  more  monomials^  take 
the  j)roduct  of  the  greatest  common  divisor  of  their  numerical  coeffi- 
cients and  each  letter  raised  to  the  lowest  power  to  which  it  appears 
in  any  of  the  monomials. 

In  polynomials,  factor  each  expression  and  proceed  as  with  mono- 
mialSf  treating  each  factor  as  you  would  a  single  letter. 


156  ELEMENTARY  ALGEBRA 

EXERCISES 
Find  the  highest  common  factor  of : 

1.  A.a%\Qa^h\  7.   ^9  wi^t\  21  w^r^l^u. 

2.  9  a*hh\  12  a^feV.  8.   35  CWP',  -  56  CD^P\ 

3.  IBWV^^OWV^IP,  9.   24m2wy,  42  mpS  18  my^. 

4.  8  mnV,  96  mV^.  10.    12  (v  +  Q^  8  (v  +  0^- 

5.  16  M'N\  24  M^N^P".  11.   14  (^  -  i^)«,  21  {E  -  F)\ 

6.  2^Y^\f^'  12.   6(a;-l)2,  8(a;-l)(a;-2). 

13.  10(2/  +  l)^4(2/  +  l)2(2/-l). 

14.  39  (m  -  7i)2  (m  +  nf,  26  (m  -  ti)  (m  +  n)\ 

15.  A'-1,A'-^2A-S. 

16.  1-^2,  1-^,1-^. 

17.  F^  +  2  F-  15,  F^  -  4  F+  3. 

18.  a2_52^a»-6^a2-3a&  +  26«. 

19.  2ar^  + 4  a;,  4  0)3 +  120^ +  8  a;. 

20.  i22_32^  9_  J22  222-5i2  +  6. 

21.  'm;^  _  1^  ^3  _|_  j^^  ^  _j_  1^ 

22.  2a2  +  a- 6,  6a2-7a-3. 

23.  3rir2«-3riV2,  4riV  +  2riV-6riV8. 

24.  ^''-^^,  ^^  +  ^'^  +  -4B+-B^. 

25.  6m^w  +  14m27i2  +  4m7i3,4m*  +  16m''n+16mV. 

95.  Lowest  Common  Multiple.  —  A  multiple  of  a  given  expression 
is  any  expression  of  which  the  given  expression  is  a  factor.  A 
common  multiple  of  two  or  more  expressions  is  an  expression  that 
is  a  multiple  of  each.  The  lowest  common  multiple  (L.  C.  M.)  of 
two  or  more  given  expressions  is  the  expression  with  the  least 
numerical  coefficient  and  the  lowest  powers  of  its  literal  factors 
that  is  a  common  multiple  of  the  given  expressions. 


FACTORS.    MULTIPLES.    EQUATIONS  167 

Example  1.  —  Find  the  L.  C.  M.  of  24  A^B^C*  and  30  A^B^G^. 
The  least  number  that  will  contain  both  24  and  30  is  120. 
The  lowest  power  of  A  of  which  A^  and  A^  are  factors  is  A*. 
The  lowest  power  of  B  of  which  B^  and  B^  are  factors  is  B^. 
The  lowest  power  of  C  of  which  C*  and  C^  are  factors  is  C*. 
Hence,  the  L.  C.  M.  is  120  A^B^C*. 

Example  2.  —  Find  the  L.  C.  M.  oi2v^  -  iv  ■{•  2  and  3 1;2  +  3  v  -  6. 
2u2_4^  _^  2    =2(t>-  1)2. 
3t)2  +  3D-6     =3(v  + 2)(r  -  1). 
Hence,  L.  C.  M.  =  6{v  -  iy{v  -\-  2). 

To  find  the  lowest  common  multiple  of  two  or  more  monomials,  take 
the  product  of  the  least  common  multiple  of  the  numerical  coefficients^ 
and  each  letter  raised  to  the  highest  power  to  which  it  appears  in  any 
one  of  the  monomials. 

In  polynomials,  factor  each  eocpression,  and  proceed  as  with  m<h 
nomials,  treating  each  factor  as  you  would  a  single  letter. 

EXERCISES 
Find  the  lowest  common  multiple  of : 

1.  2a%Sab\  10.   {A-By,S(A-B)(Ai-By. 

2.  6  a^y^,  10  xy\  11.   n(n  -  l)(n  +  1),  2  n^n  + 1). 

3.  Stvh'u^,12wv^.  12.   t^(l-\-ty(l-t),{l+t){l-t). 

4.  S1]^N',1SMN\  13.   8^-1,8^-28  +  1. 

5.  7p(fr,  3p^q,  6p^7^.  14.   D'-1,D'-D-  2. 

6.  21  A'BC,  4.  A^BC,  6  AB^.   15.    l-y%y  +  f-2f. 

7.  25 m'n\  4.  a7nn%  10  ahnn\     16.    F^- (^,  2  F'+SPQ+Q'. 

8.  ISi?'?^,  6i2V,  4i2V.  17.   t^-\-5t-{-6,t^  +  7t-\-12. 

9.  (xi-yy,2(x  +  yy.  18.   v" -{-v  -  SO,  v^ -h  5v-6. 

19.  2A^  +  3A-2,3A'  +  7A+2. 

20.  2o^-x-10,2a^-\-x-3. 

21.  2IP  +  2N,N'-^5N-hQ- 

22.  J<^-l,]<^-k,2J<?. 


158  ELEMENTARY  ALGEBRA        - 

23.  P*-l,P'-\-3P  +  2,F'-\-P-2. 

24.  1  -  d',  1  -d%l-  d. 

25.  z^-z\2z^-4:Z,z^  +  l. 

26.  B'  +  5B-U,AB'-16B''  +  16B,SB^ 

27.  a'-b^  a^-b%a*-b\ 

28.  a:^+ab  +  ac+  be,  a^-^2  ab  +  W. 

29.  x^  —  y^,  x^  —  xy  +  XV  —  yv,  a^  -{-  xy  -\-  yv  +  aw. 

30.  .4^-^,  ^^4-2^2^2_,_^^^4_2^2^2^^ 

EQUATIONS    SOLVED    BY   FACTORING 

96.  Equations  solved  by  Factoring.  —  One  important  use  of  factor, 
ing  is  in  the  solution  of  equations  of  the  second  or  higher  degrees. 
In  solving  an  equation  by  factoring,  use  is  made  of  the  principle 
that: 

If  one  of  the  factors  of  an  expression  is  zero,  the  whole  expression 
must  be  zero. 

Example. — Solvew2+n=6. 
Transposing,  w^  -H  «  —  6  =  0. 
Factoring,  {n  -  2)  (n  +  3)  =  0. 

Now  this  equation  is  satisfied  by  any  value  of  n  that  ■will  make  either  of  the 
factors  n  —  2  or  w  +  3  zero,  since  then  their  product  must  be  zero.  Such 
values  of  n  are  obtained  by  setting  each  factor  equal  to  zero,  and  solving  the 
resulting  linear  equations. 

Setting  each  factor  equal  to  zero, 

w  —  2  =  0  ;  whence  w  =  2. 
w  +  3  =  0  ;  whence  n  =—3. 
Check. — When  n=  2,  the  equation  becomes  4  +  2  =  6. 

When  w  =  —  3,  the  equation  becomes  9  —  3  =  6. 

To  solve  an  equation  by  factoring : 

(1)  Transpose  all  terms  to  the  first  member,  Tnaking  the  second 
member  0. 

(2)  Factor  the  resulting  first  member. 

(3)  Set  each  of  the  factors  equal  to  0,  and  solve  the  resulting 
equations. 


FACTORS.    MULTIPLES.  EQUATIONS  159 

It  will  be  found  that  an  equation  of  the  second  degree  has  two 
roots,  an  equation  of  the  third  degree  three  roots,  etc. 

Note.  — Care  must  be  exercised  not  to  divide  both  members  of  an  equation 
by  an  expression  containing  the  unknown  number,  because  in  so  doing  roots 
are  lost. 

EXERCISES 
Solve: 

1.  w2-6n  +  8  =  0.  21.  10  =  3^  +  ^2^ 

2.  62_^4&  =  12.  22.  6^  =  9  +  ^-2. 

3.  a^-9  =  0.  23.  02^3^.^54 

4.  P2  =  16.  24.  22^*4-2  =  53;. 

5.  4a2-25  =  0.  25.  2i^  +  ^p^^, 

6.  9i2_49^o^  26.  5^  =  1  +  4J5. 

7.  >S2  =  64.  27.  21t«2  =  10  +  29w. 

8.  4-r2  =  0.  28.  «3-a;  =  0. 

9.  552  +  6  =  4^52+7.  29.  m3  +  2m2  =  3m. 

10.  7cZ2-28  =  0.  30.  F^  =  3F2  +  10F. 

11.  ^(^  +  4)  =  4^+49.  31.  4^  +  10^  +  4^  =  0. 

12.  c2  =  8c  +  9.  32.  a3  +  a2  =  9a  +  9. 

13.  <2  =  22  +  9<.  33.  i2'»  +  4  =  i22  +  4i?. 

14.  F2-15F=54.  34.  3/_p2  =  27p-9. 

15.  2>2  +  12i>  +  20  =  0.  35.  32  +  ar»  =  16a;  +  2a;». 

":U  jp2^6-p.  36.  iV*-5i\r2  +  4  =  0. 

p-  ■  ■  '} 

*.'i7.    Tr2+14  TF+40  =  0.  37.  v^  =  ^v\ 

;ri8.    7i2-llA  +  30  =  0.  38.  ^^  +  12^2^7^43. 

19.  iV^  +  5^=14.  39.  «^-18«2  +  81  =  0. 

20.  2/2  +  12  =  72/.  40.  2 D*  + 18  =  20 Z)^. 


160  ELEMENTARY  ALGEBRA 

41.   Explain  the  fallacy  in  the  following  reasoning,  by  which 


it  is  shown  that 

2  =  1. 

Let 

w  =  l. 

Multiplying  by  w, 

w2  =  n. 

Hence, 

n2_i  =  n-l 

Dividing  by  n  —  1, 

71  +  1  =  1. 

Substituting  1  for  n, 

2  =  1. 

42.  If  to  the  square  of  a  certain  number  five  times  the  number 
is  added,  the  sum  is  104.     Find  the  number. 

43.  Find  two  consecutive  whole  numbers  whose  product  is  182. 

44.  The  sum  of  the  squares  of  two  consecutive  odd  numbers  is 
202.     Find  the  numbers. 

45.  The  sum  of  the  squares  of  three  consecutive  numbers  is 
194.     Find  the  numbers. 

46.  The  area  of  a  triangle  is  144  sq.  in.,  and  the  altitude  is 
twice  as  long  as  the  base.     Find  the  base  and  altitude. 

47.  The  base  of  a  triangle  is  7  ft.  longer  than  the  altitude,  and 
the  area  is  60  sq.  ft.     Find  the  base  and  altitude. 

48.  The  base  of  a  right  triangle  is  3  ft.  greater  than  the .  alti- 
tude, and  the  hypotenuse  is  15  ft.  Find  the  length  of  the  base 
and  altitude. 

49.  One  base  of  a  trapezoid  is  2  in.  greater  than  the  altitude, 
and  the  other  base  is  4  in.  greater  than  the  altitude.  The  area  is 
54  sq.  in.     Find  the  bases  and  the  altitude. 

50.  The  altitude  of  a  trapezoid  equals  one  base  and  exceeds 
the  other  base  by  4  in.  The  area  is  120  sq.  in. 
Find  the  bases  and  altitude. 

51.  A  square  box  without  a  lid  and  6  in. 
deep  is  to  be  made  from  a  square  piece  of  tin 
by  cutting  out  a  square  from  each  corner, 
bending  up  the  sides  and  soldering  along 
the   edges.      The   box   is   to   hold  384  cu.  in. 

Find  how  large  to  cut  the  piece  of  tin  from  which  to  make  the 

box. 


L..iHi...J 


FACTORS,    MULTIPLES.     EQUATIONS  161 

52.  An  open  box,  to  be  4  in.  deep,  twice  as  long  as  wide,  and 
to  hold  512  cu.  in.,  is  to  be  made  from  a  rectangular  piece  of 
cardboard  by  cutting  out  a  square  from  each  corner,  bending  up 
the  sides  and  pasting  along  the  edges.  Find  how  large  to  cut 
the  cardboard  from  which  to  make  it. 

53.  A  farmer  has  a  field  60  rd.  wide  and  80  rd.  long,  which  he 
is  plowing  to  sow  with  wheat.  How  wide  a  strip  must  he  plow 
around  the  field  in  order  to  have  it  half  done  ? 

54.  Two  boys  have  a  lawn  to  mow  that  is  60  ft.  long  and  32  ft. 
wide.  The  first  boy  is  to  mow  half  of  it  by  cutting  a  strip  of 
uniform  width  around  it,  and  the  other  is  to  finish  it.  How  wide 
a  strip  must  the  first  boy  cut  ? 

55.  By  measuring  the  side  of  a  room  15  ft.  square,  a  man  found 
that  he  got  an  area  3500  sq.  in.  too  small.  Find  the  amount  of 
his  error  in  measuring  the  side  of  the  room. 

Suggestion.  —  If  e  inches  is  the  error,  show  that  e^  —  860  e  +  3500;=  0. 

56.  A  room  is  10  ft.  square.  What  error  (in  inches)  in  meas- 
uring the  side  would  make  the  computed  area  476  sq.  in.  too 
small? 

57.  In  the  room  in  Problem  56,  what  error  in  measuring  the 
side  would  make  the  computed  area  729  sq.  in.  too  large  ? 

58.  The  horse  power  of  a  gasoline  engine  is  computed  by  the 
formula  H= ,  where  H  =  horse  power,  D  =  diameter  of  cyl- 

inders  in  inches,  and  N=  number  of  cylinders.  In  an  engine 
of  one  cylinder,  find  the  diameter  of  the  cylinder  required  to 
yield  40  horse  power. 

59.  In  a  gasoline  engine  of  4  cylinders,  find  the  diameter  of  the 
cylinders  required  to  yield  160  horse  power. 

60.  From  a  group  of  n  children  two  leaders  for  a  game  may  be 
selected  in  n(7i  —  1)  ways.  If  the  number  of  ways  in  which  the 
two  leaders  may  be  chosen  is  240,  how  many  children  are  there  in 
the  group  ?     . 


162 


ELEMENTARY  ALGEBRA 


SUPPLEMENTARY  EXERCISES 


Factor : 


\/ 


1.  4^2^-8^. 


'10.   m^  —  n^ -\- mx  —  nx. 

Suggestion. — Group  and  factor 
the  first  two  terms,  then  group  and 
factor  the  last  two  terms. 


11.  V-  W  +  tV-tW. 

12.  x  —  a-\-{x—ay. 

13.  31?  —  X  —  y^  -f  y. 

14.  m^n^  —  m*  —  71^  -f  1. 


^  2.    ax""  +  6a;^ 

(/  3.     t;«  _  ti;i;»». 

1/5.   P'-'-5P^Q\ 

•  7.  5  v«2/"~^  —  10 -?;*- V-' 
{/S.  8  Jtf  2»+i  + 12  Jf  2«jv: 
/>^.   a262_(a-6)2.  15.   n*  -  r^*  -  (rj^  -  ra^)^ 

16.  Af-\-At-\-Bf+Bt+A-{-B. 

Suggestion.  —  Group  the  first,  second,  and  fifth  terms ;    also  group  the 
third,  fourth,  and  sixth  terms.     One  factor  is  a  trinomial. 

17.  2a-2h-^ad—hd-{-2c-[-cd. 

18.  ny''-\-^ny-\-by^  +  l^y  +  2n^l0. 

19.  p^-\-A:p^-\-^p  — p^q—4,pq  —  ^q. 

20.  x^  —  y^-\-z^  —  w^  —  2{xz  —  yw)'  ; 


21.  W^-\-  TF^+T^-  y\ 

22.  A^-\-A^C^-ABC^B^C-W. 

Suggestion.  —  Group  the  first  and 
last  terms  together,  and  the  other 
three  terms  together. 

23.  4  mV-(m2  4-^2  _p2)2. 

24.  a(a-\-c)-h(^-\-c)\ 

25.  w^  —  v^-\-u{\i  —  2w). 

26.  aj2n  _  2  a;«2/**  +  y2n^ 

27.  (i~«^$a''^"-j-9a''i2« 


28.  6P-9P2_i. 

Suggestion.  —  First    remove    the 
monomial  factor  —  1. 

29.  -^4_j_g^2_ig^ 

30.  -ic2 4-2^-2/^. 

31.  -4Jf2-l2il[/i\r~9iV^2 

32.  (a-}-6)2  +  7(a  +  6)+10. 

33.  (a;-2/)2_6(a;-y)-4a 

34.  ay?  ■\-{a-\-h)x -{■}). 

35.  aa;2-f  (6-a)cc-6. 


FACTORS.     MULTIPLES.    EQUATIONS  163 

36.  2/  +  (4a  +  6)2/  +  2a6.  43.  ««-/. 

««     o  TT7-2     /o        .     NTTT  .  SUGGESTION.  —  Flrst  factor  as  the 

37.  2  W- (2m-{-  n)W+  mn.    difference  of  two  squares,  then  factor 

38.  av^+(ab-^l)v  +  b.  the  factors. 

^'  44.   l-n«. 

40.  4^^-9B««.  45    ^_1 

41.  w;3«  +  v3".  46,   a«-64  6«. 

42.  1-Q^.  47.   B}^-7^. 
Solve  the  following  equations  for  a; : 

48.  a,-2-3aa;  +  2a2  =  0-  52.  a;^ - 1(;2 ^ 2 wv  +  v^. 

49.  wV-ri^  =  0.  63.  a;24.94.4a5=6a;H-a2-|-462. 

50.  4va;  =  ic2_^4^^      .  54  ar^  =  2 rx  +  63 r^. 

51.  2x^-3^  =  ^x1.  55.  a^  +  7iar^  =  62a;+62n. 

56.  Any  two  arithmetical  numbers  with  three  figures  each  may 
be  written  in  the  polynomial  forms  100  C  -\-10  B-{-  A  and  100  c 
+  10  6  -I-  a,  respectively.     Show  that  their  product  is 

10000  (7c  +  1000(C6  +  cB)  +  100(ea+B6+^c)  +  10(^6+a5)H-^a. 

From  the  form  of  this  expression  may  be  deduced  a  process, 
called  "  cross-multiplication,"  for  finding  the  product  of  two  arith- 
metical numbers  in  an  abbreviated  form.  The  process  is  appli- 
cable to  numbers  with  any  number  of  figures.  By  this  process 
practically  all  of  the  work  in  multiplica-  4-3  6 

tion  is  done  without  the  aid  of  pencil,  but 
it  requires  great  mental  concentration. 

Thus,  to  find  274  x  436  the  work  is  done 
mentally  as  follows,  and  only  the  answer  writ- 
ten :  To  get  the  ones,  take  4  x  6  =  24  and  write 
the  4  ;  to  get  the  tens,  take  2  +  4x3  +  6x7=  56  and  write  the  6 ;  to  get 
the  hundreds,  take  5  +  4x4  +  6x2  +  7x3  =  54  and  write  the  4  ;  then 
take  6  +  7  x4  +  3x2  =  39,  and  write  the  9  ;  then  take  3  +  2  x  4  =  11. 

57.  Find,  by  "  cross-multiplication,"  the  products  of :  34  x  52 ; 
61x36;  47  X54;  67x62;  52x83;  245x326;  318x241;  234 
X  571 ;    384  x  615  ;    724  X  318  ;    27  x  356    (same   as  Oi27  X  356> 


CHAPTER  IX 
FRACTIONS 

97.  Algebraic  Fractions. — Tlie  indicated  quotient  of  two  num- 
bers or  algebraic  expressions,  as  —  or  — — -^ -,  is  called  a 

n        or  -i-2ab  -{-  Ir 

fraction.  In  algebra,  as  in  arithmetic,  the  dividend  is  called  the 
numerator,  and  the  divisor  the  denominator,  of  the  fraction.  The 
numerator  and  denominator  are  called  the  terms  of  the  fraction. 

2a;  —  1 

The  fraction is  read  either  "  2  ic  —  1  divided  by 

x^  -\-x  -\-l^^  or,  more  briefly,  " 2 a;  —  1  over  x^  ■{-  x  -\- 1.'* 

98.  Signs  of  a  Fraction.  —  In  an  algebraic  fraction  there  are 
three  signs  involved:  (1)  the  sign  before  the  fraction,  which  is  the 
plus  or  minus  sign  written  (or  understood)  before  the  line  sepa- 
rating the  numerator  from  the  denominator;  (2)  the  sigyi  of  the 
numerator  ;  (3)  the  sign  of  the  denominator. 

Since  a  fraction  is  an  indicated  quotient,  it  follows  from  the 

laws  of  signs  in  division  that  i-^  and  ^^^  are  both  positive  and 

+  6  —  h 

are  equal,  and  that  ^^  and  -^t_^  are  both  negative  and  are  equal. 
-{•0  —b 

Hence,  '  ±^  ==  =1^  =- Zl^  =  -±^. 

'  +6      -b  +6  -6 

It  follows  that  to  preserve  the  algebraic  value  of  a  fraction 

(1)  If  the  signs  of  both  terms  of  the  fraction  are  changed,  the  sign 
before  the  fraction'  must  be  left  unchanged, 

164 


FRACTIONS  165 

(2)  If  the  sign  of  only  one  term  of  the  fraction  is  changed,  the  sign 
before  the  fraction  must  be  changed. 

+  4        -4  +4  -4 

If  the  numerator  or  denominator  be  a  polynomial,  its  sign  is 
changed  by  changing  the  sign  of  each  of  its  terms,  because  this  is 
equivalent  to  multiplying  the  polynomial  by  —  1. 

Thus  a;g-4a;  +  2    ^  -  a;2  +  4a;~  2  ^^      x^-4x-{-2 

*         -a;2  +  3x-l        x^-3x  +  l  x^-Sx  +  l 


EXERCISES 
Write  with  positive  numerators  and  denominators : 


'•^• 

'■^■ 

7     -2a' 

-z1^- 

..^. 

'■5|- 

-^. 

".  -;r- 

'■^. 

'■m- 

'■^ 

—  vg* 

iVrite  with  denominators  a  ■ 

-6 

: 

^vh- 

14. 

F- 

-  X 

-  a' 

15.    ^-'*. 
b  —  a 

In  each  of  the  following  expressions  write  all  of  the  fractions 
\*  ih  the  same  denominator : 

16.   _6_  +  _2_.  18.  2Mil__3^  +  l^. 

x  —  y      y  —  x  ar*  — 1       1  —  a^      l—o? 

yj        a      ,       b     __  a  +  h  -g      bat a^      a'  +  f 

•   a- 6"^  6- a      6  -  a'  '  a' -  t^      ^  _  a^  "^  a*  -  i»' 

„^    1  +  r  ^     +      1 


1  —  r      r— 1      r  —  1 


166  ELEMENTARY  ALGEBRA 

Write  for  eacli  of  the  following  an  equal  fraction  with  positive 
denominator  and  the  sign  -f  before  the  fraction : 

21.    -JL.        23.    -J#-.       25.         -1  "-       -'^ 


-A' 

/it* 

-TTT^' 

-5 
—  n 

28. 

-100* 

22.    -^.  24.    -I^.       26. 

99.  Reduction  of  Fractions  to  Lowest  Terms.  —  As  in  arithmetic, 
to  change  a  fraction  to  lowest  terms  is  to  change  it  to  a  fraction 
with  equal  value,  but  whose  numerator  and  denominator  have  no 
common  factor. 

Thus,  6^2x^  =  5.     1=?     1^=.?    i^  =  ? 

8      2  X  4      4       18  24  100 

The  principle  involved  in  the  process  is  that : 

Dividing  both  terms  of  a  fraction  by  the  same  expression  does  not 
change  the  value  of  the  fraction.  .  :, 

Example  1.  —  Reduce  — - —  to  lowest  terms. 
ISa^b* 

Dividing  both  terms  by  6  a%^^  their  highest  common  factor, 
12  a^h^  ^  2  62 
18a3&*      3  a* 

Example  2.  — Reduce  ^-i — ^~     to  lowest  terms. 

ic2  +  5  X  +  6 

Factoring  both  terms,  and  dividing  by  oj  +  3,  their  H.  C.  F., 

x2  +  2a;-3^  (a;  +  3)  (a;  -  1)  ^x-\^ 
ic2  4.6x  +  6      (ic  +  3)(x  +  2)      x  +  2 

It  is  evident  that : 

To  reduce  an  algebraic  fraction  to  its  lowest  terms,  factor  the 
numerator  and  denominator^  and  divide  both  terms  by  their  highest 
common  factor.  ^  .. 


FRACTIONS  167 

The  division  may  be  indicated  by  cancellation  of  the  commou 
factors  of  the  terms. 

Thus    ^^^  —  g?^^  _.        iKja^-^^^jm  +  n)        _        m  -ir  n 
'  am^  —  an^      jiif(wj^--7?J(m'^  +  win  +  w^)      wi'-^  +  win  +  n^ 

Note. — It  must  be  remembered  in  using  cancellation  that' only  factors 
common  to  the  numerator  and  denominator  may  be  cancelled.  Terms  in 
polynomial  numerators  and  denominators  cannot  be  cancelled  —  a  common 

3  4-5 

mistake  made  by  students.    Thus,  in     "^     the  5's  cannot  be  cancelled,  be- 

4  +  5 

cause  the  fraction  equals  |,  and  if  the  5's  were  cancelled,  it  would  become  |. 
Similarly,  in  ^  "^     no  cancellation  is  possible. 


EXERCISES 

Reduce  to  lowest  terms : 

1.    1^.  6.    ?5.  11.    «5.  16.    ^^. 

12  96  ac  10  yz 


—  •  7.    — •  12.    •  17. 

12  54  6ar^  12  wv' 


1^  8.   -I?-.  13.    i^.  18.     '^'"'^ 


18  100  6m?i2  21  i^st 

4     ?1.                   9  ?40,  12j)V  ,9     8^^ 

*    28                      *  800  *    16j9V  *    36  v^ 

6.    i?.                 10.  -5^.  15.   A^.  20.    i^. 

20  720  Qa^hd  irl^ 

4.7rR(E-\-H)  A'-AB^  4  7t^4-47i  +  l 

STrH^iE  +  H)           *  ^2^2^J3+^2  •  4^2-1 

22.          ^'-^ 26.  ^'-^     >              30  ^^^^ 

a2  +  2a6  +  i>'  2By  +  2y  a^-b^ 


A'-{-2AB-{-B^ 

R'-l 

2Ry  +  2y 

^2  +  2^-15 

f-t-  6 

l-p« 

om--xm^  ^2  +  2  ^ - 15  l-a; 

24.    ^^.  28.    _1^I^!_.  32.    i^^I^. 


168  ELEMENTARY  ALGEBRA 

33     F^  +  8F+12  M'-M'N^  r,%-nr,\ 

34.    ^^±^.  36.         ^-^       .  38.  ^'-^' 


39     ^-^-2.  42       6a^  +  7a;-5    . 

•  4-i>2  •   3a^  +  17a;  +  20 

^^     ac-6c-ad4-5cg^  ^3      10 a" -S3  at-^7  t^ 

•  ac4-ad-&c-6d'  *   12  a^  -  52  a«  +  35  i«* 

B^-Br-B  +  r  l  +  P' 

100.  Reduction  of  Fractions  to  Mixed  Expressions.  —  An  algebraic 
expression  containing  no  fractions  with  literal  denominators  is  an 
integral  expression ;  an  expression  consisting  of  one  or  more  such 
fractions  is  a  fractional  expression;  and  an  expression  consisting 
of  an  integral  part  and  a  fractional  part  is  a  mixed  expression. 

/v. 1  A  A 

Thus,  x^ ■  and ^=—  +  A^  —  1  are  mixed  expressions. 

x2  +  l  ^-lu4  +  l 

Note.  —  In  arithmetic  the  mixed  number  4|  means  4  +  f ,  the  sign  +  being 
omitted.  In  algebra  the  sign  between  the  integral  and  fractional  parts  of  a 
mixed  expression  must  never  be  omitted.  Thus,  x  +  -  may  not  be  written  x  - , 
because  the  latter  implies  multiplication. 

An  algebraic  fraction  whose  numerator  is  of  as  high  degree  as 
the  denominator,  or  of  higher  degree,  may  often  be  reduced  to  a 
mixed  expression.  The  process  is  similar  to  that  in  arithmetic 
of  reducing  an  improper  fraction  to  a  mixed  number. 

Example  1.  —  Reduce  ^^-^-  to  a  mixed  number. 
Dividing  731  by  46  gives  a  quotient  15  and  remainder  41. 
Hence,  3^%^  =  15^. 

Example  2.  —  Reduce  ^-i — ^'~     to  a  mixed  expression. 

JD  +  3 

Dividing  x^  +  6x  —  Qlaj  x  +  S  gives  a  quotient  oj  +  2  and  remainder  —  15. 

15 


Hence,  ^'  +  ^^-Q  =  x  +  2  + 

'        x+3  x+3 

15 

This  answer  may  be  written  x  +  2  —  ■ 

x  +  3 


FRACTIONS  169 

To  reduce  a  fraction  to  a  mixed  expression,  divide  the  numerator 
by  Uie  denominator.  The  mixed  expression  equals  the  quotient  plus 
the  fraction  whose  numerator  is  the  remainder  and  denominator  the 
denominator  of  the  given  fraction. 

EXERCISES 
Reduce  to  mixed  numbers : 


1.    i.               3.    V.              5.    -1^. 

7-    W-                9-   ^4W- 

2.    ^.             4.    If.              6.    ^. 

8-  m-      10-  ^w- 

Reduce  to  mixed  expressions : 

.1.    ^  +  1.                     16.       P'    . 

„^    ^A'+S'AB+iB? 

3A 

-2B 

22 

v" 

-fi 

V 

+  t 

23. 

V^ 

-Sn' 

'-\-Sn- 

-1 

n  +  1 

S'- 

-16 

jS- 

-2 

^-f  4^  +  3 

x-1  P-3 

2^-4       *       *     M+n' 

^^     7i^-5n  +  l.         ig     r^  +  3r^  +  5r4-2 
w-2  *  r  +  1 

14.    -1^^.  19.    ^+^.  24. 

16.        ^•^  +  ^     .  20.    5^+«,\  25. 

2«2_^4-2  a-2  E'-'SE 

101.  Reduction  of  Fractions  to  Higher  Terms. — It  sometimes  is 
necessary  to  change  a  fraction  to  an  equivalent  fraction  with 
denominator  of  higher  degree.     The  principle  involved  is  that: 

If  both  terms  of  a  fraction  are  midtiplied  by  the  same  expression, 
the  value  of  the  fraction  is  not  changed. 

The  process  is  the  same  as  that  in  arithmetic  of  reducing  a 
fraction  to  higher  terms. 

Example  1.  —  Reduce  |  to  48ths. 

Dividing  48  by  8  gives  6.     Heuce,  multiplying  both  terms  by  6, 

i  =  ih 


170  ELEMENTARY  ALGEBRA 

Example  2.  —  Change  -^-^t —  to  a  fraction  with  denominator  6  a^+aft  — 6*. 
2  a  +  6 

Dividing  6  a^  +  a6  —  6^  by  2  a  +  6  gives  3  a  —  6.  Hence,  multiplying 
both  terms  by  3  a  —  6  gives 

a  +  h   _  3  qi'  +  2  a5  -  62 
2  a  +  &       6  a2  +  a6  -  62  * 

Divide  the  required  denominator  by  the  given  denominator,  and 
multiply  both  terms  of  the  fraction  by  the  quotient. 

Note.  — The  division  may  be  quickly  performed  by  factoring  the  required 
denominator  and  cancelling  from  it  the  factors  of  the  given  denominator. 
Thus,  in  Example  2,  the  required  denominator6  a^+ab—b^=  (2  a  +  6)  (3  a--6). 
Cancelling  2  a  +  6,  the  quotient  is  3  a  —  6. 


EXERCISES 

1.  Change  f  to  26ths.  3.   Change  ^  to  64ths. 

2.  Change  ^  to  56ths.  4.    Change  ^  to  64ths. 

5.  Change  ^  to  126ths. 

6.  Change  each  of  the  following  to  72(is : 

JL  3.  5 


If 


4>     ■§">     jf    T2">     a* 


7.  Change  — -  to  a  fraction  with  denominator  a^6V. 

b^c 

8.  Change to  a  fraction  with  denominator  x^  —  1. 

x—1 

9.  Change  -— to  a  fraction  with  denominator  M^  —  N^, 

M  -\-  N 

10.  Change  — — -  to  a  fraction  with  denominator  R^-{-3Ii-^2, 

M  -{-1 

11.  Change  ^^-^  to  a  fraction  with  denominator  v'  —  ^. 

v  —  t 

12.  Change -—     to     a     fraction     with     denominator 

2x-\-6y 

6V4-7icy-202/2. 


FRACTIONS  171 

XS.   Change  each  of  the  following  to  a  fraction  with  denom- 
inator a^h  —  aWi  ^  ^         ^      J 


a+h     a—b     a     b 
14.   Change  each  of  the  following  to  a  fraction  with  denominator 

a;-?/'    x  +  2/'    {x-yf'    (x  +  yf 

102.   Reduction  of  Fractions  to  a  Common  Denominator.  —  It  is 

clear  that  by  the  process  of  §  101  two  or  more  fractions  may  be 
changed  to  equivalent  fractions  having  for  the  denominator 
of  each  the  lowest  common  multiple  of  all  of  the  denominators 
of  the  given  fractions.  This  lowest  common  multiple  of  the 
denominators  of  the  given  fractions  is  called  the  lowest  common 
denominator  (L.  C.  D.)  of  the  fractions. 

Example.  —  Reduce  ^  "^  '   and ^^ —  to  equivalent  fractions  having 

a;2_i         a;2_4a;_5 

the  lowest  common  denominator. 

a;2-l  =  (x  +  l)(a;-l). 

a;2  -  4  a;  -  6  =  (x  +  1)  (a;  -  5). 

Hence,  L.  C.  D.  =  (x  +  l)(x  -  \){x  -  5). 

Cancelling  the  factors  of  the  first  denominator  that  are  found  in  the  L.  C.  T)., 
it  is  seen  that  the  terms  of  the  first  fraction  must  be  multiplied  by  a;  —  5. 
Similarly,  the  terms  of  the  second  fraction  must  be  multiplied  by  x  —  1. 

Hence,  ^±1  =  {^-^){^^^)  or      ^^-^^-^^     .. 

a;2-l      (x-6)(x2-l)        x^-Sx^-x  +  S' 

and  1-x      ^       (x-DCl^x)        ^^    -^2^2x^1 

x^^^x-b      (X- l)(x2-4x- 6)        x8-5x2-x4-6 


EXERCISES 

Reduce  to  equivalent  fractions   having  the  least  common  de- 
nominator : 

1-  h  i.  i        3     I,  I,  f  5.    I,  f,  ^\.         7.    J,  I,  i^,  ^. 

2-  i.  *,  f       4.   h  4,  «.       6.   ■^,  A.  i-       8.    A,    A.   h   h 


172  ELEMENTARY  ALGEBRA 

Reduce  to  equivalent  fractions  having  the  lowest  common  de- 
nominator : 

9  A  J_  A.  12.  i-,  ^,  y-. 

'    Sx    2x^'    6x  '    xy   yz   xz 

10.    1.,  A,    1  13.        ^  ^  1 


Sf'  6^'  4t*'  '    A  +  1'  A-1'  A^-1' 


11.    :rT^,     TT— T-,      .     ..  >  14 


36V    2aV    4a262-  *   n^  +  3n  +  2'  2712  +  571 +2 

15  ^-1  F+3 

F'  +  5F+4'   F2+  F-12* 

16  2ig  +  7  3ig-4 
•    6i22  +  13i2-5'    12i22-13jK  +  3* 


17.       1  ^  1 


18. 


a  -  6'    a  +  6'    a'  +  6'* 
2  c  2c2 


c  +  r  (c  +  if  (c  +  i/ 


V     i^-u  —  v^'    w^  —  v^ 

X  y  a? 

x—y    2x-2y     6(a^  —  f)' 


21  P-^  i>  +  l  i)  +  2      . 


1  1  1 


eS^  +  S-2\  15S^-\-10S'    12S'-6JS 


FRACTIONS  173 

103.  Addition  and  Subtraction  of  Fractions.  —  Algebraic  fractions 
are  added  or  subtracted  by  the  same  rules  as  fractions  are  added 
or  subtracted  in  arithmetic. 

Example  1.  — Add  |,  |,  and  |. 

Reducing  all  of  the  fractions  to  the  least  common  denominator, 

l  +  l  +  i  =  M  +  ^  +  H- 

Adding  the  numerators  and  placing  the  sum  over  the  common  denominator, 

M  +  3^\  +  il  =  M  or  n- 

^Example  2.— Add  ^^  +  \  x^^l    and  -. 
x-\      x+\  X 

Reducing  the  fractions  to  their  lowest  common  denominator, 
2a;  +  l  ,  x-2      3  _  2x^  +  ^x^  -\-x  _^  x«-3a;2  4-2x      ^x^-Z 

X—  1  X-\-\        X  X^  —  X  X^  —  X  3^  —x' 

Adding  the  numerators,  this  becomes ^     — ^^    * 

Example  3.  —  Perform  the  indicated  addition  and  subtraction  in 
5.1  3 


+ 


a2  +  5  a  +  4      a^  -  a  -  2      a'-'  +  2  a  —  8 
1  3 


a2  +  5a  +  4      a'^-a-2      a^  +  2a-B 
5  .  1 


(a  +  l)(o  +  4)       (a-2)(a  +  l)      (a+4)(a-2) 

5a-  10 ,  q  +  4 8a  +  8 

(a  +  l)(a4-4)(a-2)       («  +  !)(«  + 4)(a -2)      (a  +  l)(a  +  4)(a- 2) 

3a-9  ^„  3a-9 


(a  +  l)(a  +  4)(a-2)        a^  +  Sa'^-Ga-S 

After  reducing  the  fractions  to  a  common  denominator,  the  first  two  nu- 
merators are  added,  and  the  third  one  subtracted  from  their  sum. 

To  add  or  subtract  algebraic  fractions,  reduce  them  to  their  lowest 
common  denominator,  then  find  the  sum  or  difference  of  their  numera- 
tors, and  place  the  result  over  the  common  denominator. 


174  ELEMENTARY  ALGEBRA 

EXERCISES 

Simplify  by  adding  or  subtracting,  as  indicated : 
1-   t  +  f         3.   A  +  A-f-      5.  f-i  +  i.  7.   f-A-*- 

2.  i-A-    4.  A+i-A-    6-  A-A+A-    8.  H  +  |-«- 

-    3a  ,  5a  ,  a        ,,    3j)  ,  ■»  ,„    a     c 

*•  T+T+3-      "•  20  +  r  ^^'  b~d 

15.  £  +  1.  +  ^. 

2/2!      Qsz     xy 

16.  !L±l  +  J-  +  2"-l. 

1,.  i+JL+i. 
n?'2    ^1^3    t\n 

18       3         2TrR-\-H     H+B 
'   4.7rB  27r2?2  7ri22    * 

19.  -_^  +  _X_.  23.    -^  '^  ^ 
x^y     x-y 

20.  -^^ -^ —  24. 

1  +  a     1  —  a 

21.  m  +  n_^m-n^  2^ 
m  —  n     m-\-n 

22.  3^Zll_!Htl.  26. 
v  +  1     v-1  a2+6a  +  5     tt2  +  2a-15 

27  ^-2^  M-h4:N 

3P-{-5MN+4:JSf''     M''-MN-2N^' 

««  1  1 


S^-s 

S2  +  S       5 

205  +  1 
x-2 

.  +  2. 
2a;-l 

Sv 

^      1    ^ 

tov 

w-}-v 

to  — v     2<;^ 

-ir' 

1 

1 

2b^-\-5bc-Sc'     4  62_i3  5c  +  3c« 


FRACTIONS 


175 


29. 


80. 


2TrR 


2'rrR 


rH 


ttR  +  ttH      ttR 

3 2 

4 


31.   ^  +  ^  + 


a  +  l 


1      1-a? 


Suggestion.  — In  problems  such  as  this  it  is  best  to  arrange  the  terms  of 
all  denominators  according  to  either  the  descending  or  ascending  powers  of 
some  letter.  In  order  to  have  the  first  term  of  the  denominator  of  each  frac- 
tion positive  when  rearranged,  it  may  be  necessary  to  change  the  signs  of 


the  terms  of  the  fraction,  as  in  § 
4 


Thus,  the  last  term  in  this  problem 


may  be  changed  to  + 


a2-l 


32. 


33. 


P-1 


V 


2uv 


1_P2 
Sv 

U-\-V       U  —  V       'l^—V? 

36. 


84. 


35. 


+  T- 


2x 


l+o;     1-ar^     o^-l 
Mil 


M^-N^     M+N    N-M 


3-f-2y     16y-y«     2-3y. 
2-3/         f-^         2  +  y 


87. 


38. 


89. 


R 


R 


R' 


1+R     1-R 

b^  2  6 


R'^1 
.      1 


1-6'^ 


62  +  6  +  1      6-1 

x-^2a     x  —  2  a         4a6 
2b-x     2b  +  x     x'-Ab*' 


40. 


+ 


a—b     a+6     a— 26     a+26 


Suggestion.  —  In  certain  cases,  such  as  this,  it  is  best  to  combine  only 
part  of  the  fractions  at  a  time.  It  avoids  long  multiplications.  Here,  com- 
bine the  first  and  second  fractions,  then  the  third  and  fourth,  then  the  two 
results  obtained. 


176  ELEMENTARY  ALGEBRA 


41.  -A^+    1         1         1 


42. 


w-i-l     wH-3     w  — 1     n  — 3 

_j^ 1_  ,  _J_ L., 

B-r     B  +  r     S-s     S  +  s 


43.^4      ^  '  ^ 


44. 


y—6     y+1     y+6     y—1 
Jl 11  1 


2^+1       ^-3       3^-2 


104.  Reduction  of  Mixed  Expressions  to  Fractions.  —  A  mixed 
expression  may  be  changed  to  a  fraction  by  the  process  of  §  103. 
Compare  the  process  as  illustrated  in  the  following  example  to 
that  of  reducing  an  arithmetical  mixed  number  to  an  improper 
fraction. 

Example.  —  Reduce  m  +  n -^ —  to  a  fraction. 

m—  n 

Writing  the  integral  part  as  a  fraction  with  the  denominator  1, 

m  +  n-i =  — ; 1 

m  —  n  1  m  —  n 


m—  n       m  —  n 
w2 


m  —  n 


To  reduce  a  mixed  expression  to  a  fraction,  write  the  integral  part 
as  a  fraction  with  the  denominator  1,  and  proceed  as  in  addition  and 
subtraction  effractions. 


Reduce  to  a  fraction : 


FRACTIONS  177 

EXERCISES 


1.  4.        ^     4-2.  7.    ir4-  — < 


B  -        m^         ..  «  r^r^ 


%,  B  +  ^r^^*  5.    — ^^^: m.  8.   ri- 


1  +  5  m  —  n  '•i  +  ^2 

3.    r  +  1 -•  6.    w;  +  5h — •         9.    -  +  a-&. 

r  —  1  w  —6  a  —  o 

10.  5-^G  14.    v^-Il^-V+1. 


11.   l-«4    2.V' 


1+2/      1-2/ 


115     ^  ■      n^  —  2  -u^         n^i;  +  uv^      » 
^2  _  ^^  4_  ^2        u^^v^ 


12.  5  +  -^  +  -^.  IG.    _£!_  +  2  +  -^. 

13.  a^  +  62_a!zi|!_«5.       17.    l  +  ^  +  |. 

a  +  6  B^     B 

105.  Multiplication  of  Fractions. — The  product  of  two  or  more 
algebraic  fractions  is  obtained  by  the  same  process  as  the  product 
of  two  arithmetical  fractions. 

Thus   2^7_2x7_14.3^29_  3x2x9  _  54  . 
inus,  3  X  9  -  3  ^  9  -  27  '  6  >^  7  ^  11  -  5  X  7  X  11  "  385 

Similarly,  ^x^  =  ^;  ^^±1  ^^J^^ll  =  (jL±^)(^JL:zM. 
b      n       6»a;  +  5       a;-4         (x  +  5)(x  — 4) 

The  product  of  two  or  more  fractions  is  the  fraction  whose  numer- 
ator is  the  product  of  their  numerators  and  denominator  the  product 
of  their  denominators. 

As  in  arithmetical  fractions,  the  product  often  may  be  reduced 
to  lower  terms.     In  such  a  case,  work  may  be  saved  by  factoring 


178  ELEMENTARY  ALGEBRA 

the  terms  of  the  fractions,  and  cancelling  any  factor  of  any  numer- 
ator  by  a  like  factor  of  any  denominator,  before  performing  the 
multiplications. 

Example.  -  Find  the  product  of  2  a^  +  a6  -  6^  ^^^  Sa^^ab-2b^  , 

2  a2  -  a6  -  62  2  a^  +  5  a6  -  3  62 

2a^  +  ab-b-2^^   Sa^-ab-2b^   ^(a  +  bXJ^sr'^      Cij^^^CSa  +  2b) 
2  a2- aft -62     2a2  +  5a6-362      (2  a  +  6)  j^.^-^      ^2jl^^^  {a  +  S  b) 

_(a  +  b)(Sa  +  2b) 
(2a  +  6)(a  +  3  6) 

_3a2  +  5a6  +  262^ 
2a2  +  7a6  +  362* 


Find  the  product  of : 


EXERCISES 


1. 

|xf 

4. 

i  X  i  X  f 

7.  ixfxll. 

2. 

*x|. 

5. 

fxtfxf. 

8.   f  X  J  X  «. 

3. 

ixif. 

6. 

«  X  -jV  X  |. 

9-   iXAxfi- 

10. 

14. 

8r          g 

IS        ^     "20* 

11. 

16. 

4^3- 

3d      ay 

12. 

2<c      9/ 

16. 

?x^ 

20.    2  X  l!. 

13. 

I  ^5' 

17. 

5v      w 

21-    SX^X^. 
6^      cd      a' 

22      ^  X  ■^^^  X 

2t' 
9 

25.    ^ 

+  1 

+  2 

x*  +  2 

a^      a  +  6 

-i 

'  +  6' '   a  —  6      a  +  6' 

24.    ^x^-^. 

''■?i-: 

»^      m-\-  2n 

2t-1^2s  +  3^s  +  3 


29. 


5  +  3         t 

32. 
33. 
34. 
35. 
86. 
87. 
38. 


1       2t-l 

A-1' 

ay^^2x-S 


FRACTIONS 
30 


179 


x  +  2y       a^  —  f' 


X 


v  +  3  v  +  4  * 


iB24.3a;-10      ^  +  lx-{-12 

r'-Sr  r  +  2 

r2  +  8r+12      r«  +  r^  +  / 

6  m^  —  5  mw  —  4  w^         4  m*  —  mn  —  3  n* 


4m^  —  17m?t  —  15?i^      6m^  +  7mn-H2w' 
F«  +  6F+9  ^    F-4 


16-8F4-7^      9-F* 

6  4-  c      g^  +  a6  +  qc  +  &c 
a  +  h      h^  •{•  he  +  bd  +  cd 

l-/^^y^  +  3y  +  2^^      1 

y  +  1      y*H-y  +  l      2/-1 


39.   3f,  iV,  and  R  are  the  areas  of  the  rectangles  with  the  dimen- 
sions shown  in  the  figures.     It  is  known  from  Geometry  that 

f=  I;  and  that  4=  " 
B      b'  N     a' 


M 

d 

Q 
a6 

R 

N 

I? 

Show  by  multiplication  that  —  = 


N      a'6'* 


tSD  ELEMENTARY  ALGEBRA 

106.  Multiplication  of  Mixed  and  Fractional  Expressions.  —  Mixed 
and  fractional  expressions  to  be  multiplied  should  first  be  reduced 
to  fractions. 

Example.  —  Multiply  ^  -  1  by  —^ —  +  1. 


^^2  ]\Sn-v        I  ^2  3^ 


=  C3  w ~  ■;;) (3  71  +  ^)  ^      3n 
v^  3  w  —  « 

9  w2  +  3  WW 


EXERCISES 
Find  the  product  of : 


I      V^  K    f-^y^  +  4    ■ 


n   M  + 


FRACTIONS  181 

14    ifAnl^fJi^-JJS 
2\sH  +  sfJ\s-t      s-{-tJ 

\P-q    p  +  q    pj\  ^pq  J 

107.    Multiplication  of  Integral  Expressions  and  Fractions.  —  In 

the  multiplication  of   a  fraction  by  an  integral  expression,  the 
latter  may  first  be  written  as  a  fraction  with  the  denominator  1, 
and  then  the  product  found  as  in  the  multiplication  of  fractions. 
Thus,  12  X  ^5  =  -V  X  ^\  =  ^. 

And    (n^-4)(!!^±AJL^)=Vl^^V^J^IL^  =  et^, 
^  ^\n2-n-6  J  1  n2-n-6 

It  frequently  is  found  necessary,  as  in  solving  equations  con- 
taining fractions,  to  multiply  a  fractional  expression  by  the 
L.  C.  D.  of  the  fractions  involved. 

Example.  —  Multiply — 1 —  by  the  lowest  common  de- 

a^—1      a  — I      a  +  1 

nominator  of  all  three  fractions. 

The  L.  C.  D.  is  a^  —  1.     Multiplying  by  this  gives 


a2- 

1 

1  ^    2a2     ,  a2-l  ^,      a 

=  2  a^  +  ««  +  a  -  a^  +  a 
=:2aa  +  2a. 

a2-l         a 
1         o  +  l 

Multiply : 

EXERCISES 

.,   3x|. 

8    9X^^. 
8.   9X-. 

••  '-i 

.    „«g. 

4.   24xA. 

6.   A'X^' 

182  ELEMENTARY  ALGEBRA 

s^  20  w  2'jrR 

10.  (n^^^)(t±A2Lz:^\         12.  (^-i)f_?l_\ 

IX.   (,iJ=-.^^(|±|).  ^  13.   (2^-8)(^). 

Multiply  by  the  lowest  common  denominator : 
a+2      a-2 


20. 

3      3g- 

-7 

2 
"3 

21. 

2Z)  +  4 
32)-5 

5 
2* 

17     _2i_._3^_j!_ 

•  2/_3^y  +  3     2/'-9 

,g     _JL L_4..J_  22  37^  +  5  3Jg^  +  5ig-4 

•  -^  +  1      m-1      m  +  2  '  4i2-3  4i22_3jRH-2 

19     8      P^_P+1.  23  2A;  +  3  4A:H-5. 

•  p'^P  +  3     P-1  •  3A;-4  6A;-1 


25. 


ar^  +  a^  +  2/^     a-y       a^-2/* 
2Jf        l-3f.  7 


2Jf+3     2-Jf  2Jf2-j|[f~6 


108.  Division  of  Fractions.  —  The  quotient  of  two  algebraic 
fractions  is  obtained  by  the  same  process  as  the  quotient  of  two 
arithmetical  fractions ;  i.e.. 

To  find  the  quotient  of  two  algebraic  fractions,  multiply  the  divi' 
dend  by  the  divisor  inverted. 


FRACTIONS  183 

This  follows  from  the  principle  that  the  product  of  the  quotient 
and  divisor  must  equal  the  dividend.     If  -  and  -  represent  any 

two  fractions,  ^  -«--  =  -  x  -,  because  7  X  -  multiplied  bv  ^  gives  - 
babe  be  a  b 

by  cancellation. 

ExAMPLK—Divide  A'-^^B  +  3&  ^    A-SB^ 
A^ -\- '2  AB -{- B^  A  +  B 

A^-^AB  +  3B^  .  A-SB^(A-SB)(A-B)  ^   A  +  B 
A^  +  2AB  +  B^'A  +  B  (A  +  B)'^  A-3B 

^A-B 

a  +  b' 

Note. — If  a  dividend  or  divisor  is  a  mixed  or  fractional  expression,  it 
may  be  changed  to  a  fraction,  and  the  division  performed  by  the  above 
process. 

EXERCISES 
Perform  the  indicated  divisions : 

1.  ?-.5. 

3     6 

^'    8^4  ""'   ^- 

o     2     4 

4.    1-1^1.  S. 

12     6 

13    3^^2^  17     QM^-2M 

*     bz"    '  lOzw  '       4-3/2       •  2-i-M 

-.     20H\4.  HL  ,^     TF^-F*.      W-V 


15  .  3 

16  •  8* 

b 

5 
"6* 

hi- 

81 
100 

-fi 

9. 

2     4 

10. 

3^72 

,  67J« 

4Z>2 

•  2Z)» 

11. 

9f 

2v 
3 

12. 

8Tr 
9P  • 

"  3  * 

21jL^      3^2  TF+2F     3Tr+6F 

15     J6riV__^12_»^  ^g     a^^-lSx  +  SO  .  ar^-15a;  +  56 

16.    «±l-^£l=i.  20.  ^-1         ,ig^-12ie  +  35 

a  a»  i2*-3i?-10       i22  +  3ie  +  2 


184  ELEMENTARY  ALGEBRA 

21.    -1^„^JL-.  26.  ^-1  -        ^'-^ 


a^  +  t^     a  +  «  3ri2  +  8n-f-5     w«  +  4n  +  3 


22.    ^'  +  c^  .  36^  +  3c^  ^^^    a^-9y^  .  x^-^xy-6f 


p'—pq  +  q'  i>'  +  $ 


109.  Complex  Fractions.  —  Fractions  whose  numerators,  denomi- 
nators, or  both,  contain  fractions  are  called  complex  fractions. 

A    «  1  +  1 

d      b           W                           t 
Thus,—,   -,    -,    and   are  complex  fractions, 

d        WV  ""7 

A  complex  fraction  is  said  to  be  simplified  when  it  is  reduced 
to  an  equivalent  fraction  whose  terms  are  integral  expressions,  or 
to  an  integral  expression.  Since  a  complex  fraction  may  be  con- 
sidered an  indicated  quotient,  it  may  be  simplified  by  dividing  the 
numerator  by  the  denominator. 


Example.  —  Simplify 


A-l 


-i 


A         A         A^-^^Aj^^i, 


1+JL     ^  +  1         A         A  +  l 
A         A 


FRA  CTIONS  185 

EXERCISES 

Simplify : 

1.    |.          7.   ^.                 l  +  #                   A-l  +  --^„ 
I  1+1  12.    ^^.  17.    ±Z^. 


..    ^ 


w 


i?-l  .1-2+     ^ 


iV^  A-6 


13.    2+1.  18.    _£. 

£lzii  1+i 

_  2  « 

3*' 
4     8i.  ^^^  14.    H 

H+R 


m  +  n 

8. 

a; 

m?  —  v? 

9. 

y 

ay  +  b' 

a 
1 

10. 

n-H 

1 

u 

11. 

u  —  v 

«» 

15. 


x-y 

x-^y 

19. 

X 

y 

<»-y^ 

x-\-y 

y 

X 

R 

1-R 

20. 

1  +  /? 

'      R 

R 

1-R. 

1  +  i?        R 


X    ,1  n*    ^«* 


a*         ^,      w  — v  ^^     1  +  a;  «,     ^'2*     ^i* 


6.    -.        11.    5—-         16.    -~ 21. 

1  +  a; 


+  a;  7*1  +  -^ 


SUPPLEMENTARY  EXERCISES 

Note.  —  In  some  problems  in  the  addition  or  subtraction  of  tractions, 
such  as  the  following,  it  is  advisable  to  change  the  signs  of  factors  of  one  or 
more  denominators  before  finding  the  lowest  common  denominator.  In  this 
change  of  signs  use  is  made  of  the  principle  that,  if  the  sign  of  one  factor  of 
a  product  is  changed,  the  sign  of  the  product  is  changed ;  and  if  the  signs  of 
two  factors  of  a  product  are  changed,  the  sign  of  the  product  is  not  changed. 

Thus,  (a  -  2) (a  —  4)  =  a^  —  6  a  +  8.  If  the  sign  of  a  —  2  is  changed,  by 
changing  the  signs  of  its  terms,  we  get  (2  —  a)  (a  —  4)  =  —  a^  +  6  a  —  8.  If 
the  signs  of  both  factors  are  changed,  we  get  (2  —  a)  (4  —  a)  =  a^  —  6  a  +  8. 


186  ELEMENTARY  ALGEBRA 

Perform  the  indicated  additions  and  subtractions : 
1.1.1 


1. 


{A-B){A^C)      (B-C){B-A)     (C-A){C-B) 


Solution.  —  Writing  all  factors  of  denominators  as  A  —  B.,  B—  C,  or 
C-A, 

+ 


iA-B){A-'C)      {B-C){B-A)      {C-A){C-B) 


QA-B)iG-A)      {B-G){A-B)      {C-A){B-C) 

-\(B-C)  -.i(C-A)^ 

iA-B)(B-  G){G  -  A)'^  {A-  B){B  -  C){C  -  A) 

-HA-B) 


{A-B)(,B-G)iG-A) 
__^B-\-  G-  G  +  A-  A+B_^ 
(A-^B){B-  G)iG-A) 


,2.  1 1^+___1__ 

(x-yXx-z)      {y-x){y-z)      {x-z){z-yy 

3    mn J np pm 

{p  —  m)(p  —  n)     (m  —  n)(m—p)      (n~-p)(n  —  my 

4.    "^ +- V. + ^ . 


n+rg  ^  rg  +  y-g  ^  rg  +  n 


6. 


(a-6)(a-c)      (6-c)(6-a)^(c-a)(c-6)" 


(u  —  v){w  —  u)      (v  —  w){u  —  v)      (w;  —  u)  {v  —  w)* 


8.    •: TT^: ^-1 


FRA  CTIONS  187 

1.1 


(x^l)(x-2)      (2-x){S-x)      (a;-l)(3-a?) 


9  e  3  1 


Note.  — In  the  following  problems  invert  divisors,  and  find  the  products 
of  all  factors  at  one  step  by  cancellation. 


Simplify : 


10.   ^x?^^-^.  11    ^-^ 


12    n  —  1      n^-{-3n-10  .     n'  +  5n 
*7i  —  2        w^  +  n  —  2       n'*  —  n  —  6 


13.   (3t.^2i;/2-^i^JJ^V^^'  +  ^^^^-^^. 
^  \       2u-\-vJ  2u  +  v 


6P'-nP-10  .  7P'4-17P-12  .    10P^-27P  +  5 
•     3P2  +  2P-5    *    6P2  4-9P-2     *  21  P^H- 23  P- 20* 

Note.  —  A  good  method  of  simplifying  a  complex  fraction  (see  §  109) 
consists  of  multiplying  both  terms  of  the  fraction  by  the  lowest  common  de- 
nominator of  all  simple  fractions  in  the  terms,  as  suggested  in  the  following 
problem. 


188  ELEMENTARY  ALGEBRA 


a-\-b  .  a  —  b 
18        ^             ^ 

Suggestion.  —  Multiplying  both  terms  by 
the  L.  C.  D.  gives 

a—b     a-^b' 
a            b 

h(a-\-b)-{-a(a-b)  _^^^ 
b^a-b)  -  aia+b) 

,„    ri  +  2     n-fS 
19.               3 

rt  +  2 

2y     Sx 

n 

2A'-AB-SB' 

20.        ^    ,. 

„„    SA'-AB-2B' 
A'  +  SAB-\-2B'' 
4:A^-5AB-{-B' 

CHAPTER  X 
FRACTIONAL  EQUATIONS.    PROBLEMS.    FORMULA 

110.  Fractional  Equations.  —  Some  problems  are  expressed  and 
solved  by  means  of  equations  in  which  the  unknown  number 
appears  in  one  or  more  denominators  of  fractions.  An  equation 
of  which  the  unknown  number  appears  in  one  or  more  denomi- 
nators is  called  a  fractional  equation. 

Thus,  ^  "^     =  — }-  5  is  a  fractional  equation. 

'  n-2      n  +  2  ^ 

111.  Clearing  of  Fractions.  —  To  solve  a  fractional  equation  it  is 
necessary  first  to  change  it  so  that  it  is  free  of  fractions.  This 
process  is  called  clearing  the  equation  of  fractions. 

It  was  found  in  §  107  that  if  a  mixed  or  fractional  expression  is 
multiplied  by  the  lowest  common  denominator  of  all  of  the  frac- 
tions involved,  the  product  is  integral.  We  know  also,  as  an  axiom, 
that  if  equal  quantities,  such  as  the  members  of  an  equation,  are 
multiplied  by  the  same  quantity,  the  products  are  equal.  These 
principles  are  used  in  clearing  an  equation  of  fractions. 

Example.  —  Solve  -^  +  — ^  =  -^  +  2. 

The  L.  C.  D.  of  all  fractions  in  the  equation  is  fi  —  1.  Multiplying  both 
members  of  the  equation  by  f-^  —  1,  by  multiplying  every  term  of  each  mem- 
ber, gives 

(''  -  ^>(^i)  +  ('^  -  'Krh)  =  ('^  -  »^(rri)  ^ '  ^''''^^ 

or  2t^  +  t^  -\- 1  =  f  -  t  -{■  2t^  ^2, 

Solving,  2f=-2, 

189 


190  ELEMENTARY  ALGEBRA 

To  dear  an  equation  of  fractions,  multiply  each  member,  by  multi- 
plying every  term  of  each  member,  by  the  lowest  common  denominator 
of  all  fractions  in  the  equation.* 


Solve: 

1. 

2  =  « 
n 

2. 

3       1 
2y     4* 

3. 

4. 

r      r      r 

1. 

6. 

x  +  1.    2 

1 
=  — . 

X 

6. 

4         7 
5c      10c 

1 
10' 

7.  ^-i+-i.=20. 
T      T      ST 

g    m  + 12^  2m +  12 
2m  3m 

2 


1- 


1. 


10.  ?=     1 


EXERCISES 

11. 

S-3 

12. 

3             2 

A:  +  1      fc  -  1* 

13. 

6              2 

2^  +  3      ^-4* 

14. 

n-3      71-5 
7i-f  9      w  +  5 

15. 

X>-1         i>-5 
2i>-5      2D-2' 

Ifi 

9               7       _^ 

5r  +  2      3r  +  4 

17. 

^      +      ^      =1. 
B-2      B+2 

18. 

1=      2/      ^      4 
y-2  '  2/  +  1 

19. 

2P+3      4P+5      ^ 
3P_4      6P-1 

0(\ 

w      _    3w;         Q 

V      V  —  1  w  -\-l      w  -{-2 

*  It  is  important  that  the  teacher  should  see  that  no  pupil  gets  a  wrong  idea  of 
the  process  of  clearing  of  fractions  through  careless  thinking.  Thus,  a  pupil  some- 
times thinks  that  clearing  an  equation  such  as  ^^^^ —  =    ~^  of  fractions  con- 

sists  of  multiplying  the  first  member  by  1  +  a;  and  the  second  by  2x  +  4,  rather 
than  each  member  by  (1  +  x)  (2  x  +  4). 


FRACTIONAL  EQUATIONS 

191 

21. 

4           5 
A- 5      A' 

-^- 

y^- 

22. 

t  +  1       2  __1 

26    c  +  3_c  +  9 

c          c  +  4* 

23. 

3y  +  62/  +  4 

26.  2»  +  l- 
4 

1          n 

^^•2/+3  + 

1                 1 

2i-3      4i2_9' 

28.    ^=: 

2  +  1 

!i^^. 

m  +  1 

2              1 

m-2      m  +  2' 

30.    A  + 

2             16 

T7-             n               TT-O                J* 

F+2       F-2      F'-4 
p     p  +  3      p  —  1* 


32     3a  +  5_3a^  +  5a-4 
4a-3      4a2-3a4-2* 

2a;  +  l  8       ^2a;~l 

'   2aj-l      4ar»-l      l  +  2» 

34.    !^!^  =  _i_  +  -  +  l 


86. 


36. 


1  —  V^        1  +  V         1  —  V 

37 4__  ^      7 

i22  +  5i2  +  6      i2  +  2      [rTS* 

1      ^  r-6  T 

T-3      T  +  3      T+3' 


87.   ^^L^  +  iL+       3 


4i-12      220      5X-15 


192  ELEMENTARY  ALGEBRA 

x  —  5  7  —  x  2x  —  15 


38. 


a^-10a;-f21      a^-Sx  +  W      ar^-12a;  +  35 


39         9^  +  17  2A-1  ^  2A  +  1 

A'-2A-A8      2A  +  12      2^-16' 


40. 


^2-2^  +  5      ^2^3^-7 


41     iV-4      N-15^2N'-10N-^1 
2^-5       N+4:         N''-N'-20  ' 


42. 


7a^  +  lla  +  4        ct4-3   ^5a  +  ll 


112.  Problems  Leading  to  Fractional  Equations. — Types  of 
problems  such  as  those  in  the  following  list  lead  to  fractional 
equations,  which  are  solved  by  the  method  of  §  111. 

EXERCISES 

1.  The  sum  of  two  numbers  is  265,  and  if  the  larger  be  divided 
by  the  smaller,  the  quotient  is  14  and  remainder  10.  Find  the 
numbers. 

Suggestion.  —  Let  n  =  the  smaller  number. 
Then  265  —  w  =  the  larger  number. 

Hence,  ?6in»  =  14  +  15. 

n  n 

2.  The  sum  of  two  numbers  is  160,  and  if  the  larger  be  divided 
by  the  smaller,  the  quotient  is  4.     Find  the  numbers. 

3.  The  difference  of  two  numbers  is  324,  and  if  the  larger  be 
divided  by  the  smaller,  the  quotient  is  3  and  remainder  80.  Find 
the  numbers. 

4.  Separate  84  into  two  parts  whose  quotient  is  f . 

5.  Separate  148  into  two  parts  such  that  if  the  larger  be  divided 
by  the  smaller,  the  quotient  is  5  and  remainder  16. 


FRACTIONAL  EQUATIONS  193 

6.  Find  where  to  divide  a  beam  18  ft.  long  into  two  parts 
such  that  their  quotient  is  Z^. 

7.  A  board  is  42  in.  long.  A  hole  is  to  be  bored  in  the  board 
so  that  its  distance  from  one  end  divided  by  its  distance  from 
the  other  end  shall  be  |.  How  far  from  one  end  must  the  hole 
be  bored  ? 

8.  A  surveyor  has  two  stakes  set  in  the  ground  150  ft.  apart. 
He  wishes  to  drive  a  third  stake  between  them  such  that  its  dis- 
tance to  the  nearer  of  the  two  divided  by  its  distance  to  the  other 
shall  be  -J.     Where  must  he  set  the  third  stake  ? 

9.  What  number  must  be  subtracted  from  each  term  of  the 
fraction  \^  in  order  that  the  result  shall  be  equal  to  ^  ? 

10.  What  number  must  be  subtracted  from  each  term  of  the 
fraction  ^  in  order  that  the  result  shall  be  equal  to  f  ? 

11.  What  number  must  be  subtracted  from  each  term  of  the 
fraction  |-J-  in  order  that  the  result  shall  be  equal  to  J  ? 

12.  What  number  must  be  added  to  each  term  of  the  fraction 
^  so  that  the  result  shall  be  equal  to  |? 

13.  What  number  must  be  added  to  each  term  of  the  fraction 
•ji^g-  so  that  the  result  shall  be  equal  to  3^? 

14.  What  number  must  be  subtracted  from  each  of  the  num- 
bers, 14,  20,  32,  and  60,  so  that  the  quotient  of  the  first  two  re- 
mainders shall  be  equal  to  the  quotient  of  the  last  two  remainders? 

15.  In  a  certain  factory  a  large  machine  can  turn  out  a  number 
of  articles  in  4  days,  and  a  smaller  sized  machine  can  turn  them 
out  in  6  days.  If  both  machines  are  operated  at  once,  how  long 
will  it  take  them  to  turn  out  the  articles  ? 

Suggestion.  —  Let  n  =  number  of  days  required  for  the  machines  to- 
gether to  turn  out  the  articles. 

Show  that  1  =  1-1-1. 

n     4     6 

16.  In  a  factory  which  manufactures  a  certain  kind  of  goods 
there  are  3  large  machines  and  5  small  ones.  To  manufacture 
the  goods  required  to  fill  an  order  would  take  either  one  of  the 


194  ELEMENTARY  ALGEBRA 

large  machines  alone  60  days,  and  either  of  the  small  machines 
alone  90  days.  If  all  8  machines  are  operated,  how  long ,  does  it 
take  to  fill  the  order  ? 

17.  In  a  city  water  system,  two  pumps  pump  the  water  into  a 
reservoir.  One  pump  would  fill  the  reservoir  in  16  hr.,  and  the 
other  would  fill  it  in  12  hr.  If  both  pumps  were  driven  at  once, 
how  long  would  it  require  to  fill  the  reservoir? 

18.  A  tank  is  fitted  with  two  pipes.  One  pipe  alone  could  fill 
the  tank  in  5  hr.,  and  the  other  pipe  alone  in  8  hr.  If  both  pipes 
were  opened  at  once,  how  long  would  it  require  to  fill  the  tank  ? 

19.  A  tank  can  be  filled  by  one  pipe  in  4  hr.  and  emptied  by 
another  in  5  hr.  If  both  pipes  are  open,  how  long  will  it  take  to 
fill  the  tank  ? 

20.  A  railroad  water  tank  can  be  pumped  full  in  4^  hr.  But 
locomotives  draw  out  and  use,  on  the  average,  a  tankful  every 
8  hr.  At  this  rate,  how  long  would  it  take  to  get  the  tank 
pumped  full? 

21.  Of  two  furnaces  in  a  school  building  one  will  burn  a  given 
quantity  of  coal  in  8  days  and  the  other  will  burn  it  in  10  days. 
If  both  furnaces  are  fired,  how  long  will  the  quantity  of  coal  last? 

22.  A  farmer  has  a  quantity  of  corn  which  he  feeds  to  his 
horses  and  hogs.  It  would  last  the  horses  alone  160  days,  and 
the  hogs  alone  60  days.  How  long  will  it  last  if  he  feeds  it  to 
both  horses  and  hogs  ? 

23.  Assuming  that  they  work  at  constant  speeds,  A  can  do  a 
piece  of  work  in  7  days  that  it  would  take  B  9  days  to  do.  If 
they  work  together,  how  long  should  it  require  them  both  to  do 
the  work  ? 

24.  A  can  do  a  piece  of  work  in  10  days ;  but  after  he  has 
worked  2  days,  B  comes  to  help  him,  and  together  they  finish  it 
in  3  more  days.  In  how  many  days  could  B  alone  have  done  the 
work  ? 

25.  John  could  remove  the  snow  from  the  sidewalk  in  30 
minutes.     His  larger  brother  James  could  do  it  in  20  minutes 


FRACTIONAL  EQUATIONS  195 

John  began  the  work,  but  later  James  took  his  place,  and  the 
snow  was  all  removed  in  25  minutes  from  the  beginning.  How 
long  did  John  work  ?  '       ' ;  ; .    -  > 

26.  A  steamboat  makes  8  miles  an  hour  against  the  wind  on  a 
journey  of  40  miles.  After  it  has  gone  15  miles,  the  wind  ceases. 
The  entire  time  consumed  on  the  journey  is  4^  hr.  Find  how 
many  miles  per  hour  the  wind  retards  the  boat. 

27.  A  train  runs  into  a  city  from  a  suburban  town,  a  distance 
of  40  mi.,  in  1  hr.  40  min.  Of  this  distance  8  mi.  are  within  the 
city  limits.  The  train  makes  50  mi.  an  hour  outside  of  the  city 
limits.     What  is  its  speed  within  the  city  limits? 

28.  Two  trains,  approaching  each  other,  leave  stations  225  miles 
apart  at  the  same  moment.  One  train  runs  5  miles  an  hour  faster 
than  the  other,  and  they  meet  at  a  point  120  miles  from  the  station 
from  which  the  faster  train  starts.    Find  the  speeds  of  the  trains. 

29.  A  motorcyclist  was  overtaken  75  miles  from  his  starting 
point  by  an  automobile  which  started  from  the  same  place  2  hours 
later,  and  traveled  10  miles  an  hour  faster.  Find  the  speed  of 
each. 

30.  A  messenger  was  started  on  a  journey  with  a  message  that 
later  was  found  to  be  wrong.  One  hour  and  20  minutes  after  he 
left,  a  second  messenger  started  to  overtake  him,  traveling  2  miles 
an  hour  faster.  The  first  messenger  was  overtaken  after  he  had 
gone  a  distance  of  10  miles.     Find  the  speed  of  each. 

31.  A  man  started  on  a  journey  of  72  miles.  After  going  42 
miles,  he  stopped  and  rested  2  hours.  Including  the  2  hours 
that  he  rested,  to  finish  the  journey  at  the  same  speed  took  him 
as  long  as  to  go  the  first  42  miles.     Find  how  fast  he  traveled. 

32.  An  aviator  flew  to  a  point  60  miles  away,  and  back,  in  5  hr. 
24  min.  His  rate  going  was  25  miles  an  hour.  What  was  his 
rate  returning  ?  , 

33.  An  aviator  started  on  a  trip  to  a  point  80  miles  away. 
After  going  30  miles  he  increased  his  speed  20  miles  an  hour, 


196 


ELEMENTARY  ALGEBRA 


and  made  the  remaining  distance  in  the  same  time  that  it  took 
him  to  fly  the  first  30  miles.  What  was  his  speed  the  first 
30  miles  ? 


34.  An  aviator  made  25  miles  an  hour  against  the  wind  on  a 
journey  of  45  miles.  After  he  had  gone  20  miles,  the  wind  ceased. 
The  entire  time  required  for  the  trip  was  1  hr.  25J  min.  What 
was  the  speed  of  the  wind  ? 


35.   In  the  National  League  of  baseball  teams,  at  one  point  of 
the  season,  Chicago  and  New  York  had  records  as  follows ; 


Won 

Lost 

Chicago 
New  York 

64 

58 

30 
32 

If  these  two  teams  were  to  play  a  final  series  of  6  games 
together,  how  many  would  Chicago  have  to  win  in  order  that  the 
quotient  of  the  number  of  games  won  and  the  number  lost  should 
be  greater  for  Chicago  than  for  New  York  ? 

Suggestion.  —  Let  x  be  the  number  Chicago  must  win  in  order  that  the 
quotients  of  the  numbers  of  games  won  and  the  numbers  lost  should  be  the 
same  for  both  teams. 

Show  that  64  +  x    ^58_^6-x, 

80+6- aj         32  +  ic 

Show  that  ic  =  1||. 

Hence,  Chicago  must  win  two  more  games  in  order  to  fin^h  ahead  of 
New  York. 

36.  The  Pittsburg  and  Philadelphia  baseball  teams  of  the 
National  League  had  records  as  follows; 


Won 

Lost 

Pittsburg 
Philadelphia 

54 

50 

34 
36 

FRACTIONAL  EQUATIONS  197 

If  these  two  teams  were  to  play  a  final  series  of  5  games  to- 
gether, how  many  would  Pittsburg  have  to  win  in  order  that  the 
quotient  of  the  number  of  games  won  and  the  number  lost  should 
be  greater  for  Pittsburg  than  for  Philadelphia? 

113.  Literal  Equations:  Formulae.  —  An  equation  that  contains 
two  or  more  letters,  such  as  the  practical  formulae  encountered  in 
the  earlier  part  of  the  book,  is  called  a  literal  equation. 

In  a  literal  equation  the  number  represented  by  any  one  of  the 
letters  involved  may  be  considered  as  the  unknown  number,  and 
its  value  solved  for.  It  is  clear  that  in  such  an  equation  the 
value  found  for  the  unknown  number  will,  in  general,  be  an  ex- 
pression containing  the  other  letters  involved  in  the  equation. 
If  a  literal  equation  is  linear  with  respect  to  the  unknown  num- 
ber, it  may  be  solved  by  the  method  of  this  chapter. 

Example.  —  Solve  x  =  ^^  +  ^  for  n. 

n 
Clearing  of  fractions,  7ix  =  na  +  6. 

Transposing,  nx  —  na  =  h. 

Uniting  terms,  (x  —  a)  n  =  6. 

6 


Dividing  by  a;  —  a,  n 


X-  a 


Note.  —  The  process  of  uniting  the  similar  terms  in  a  literal  equation,  as 
in  step  3  of  the  above  example,  is  equivalent  to  factoring  the  terms. 


EXERCISES 


1.   Solvea;  =  ^^^5±^fora. 


2.   Solve  2a-^=^  for  JB. 

X         X 


3.  Solve  a  =  ^-::^  for  a. 

x  —  1 

4.  Solve  2  +  ^-^  =  ^- ^-^  for. 


198  ELEMENTARY  ALGEBRA 

6.  Solve  -^  +  ^  =  m  -\-n  for  y. 

m     n 

7.  Solve  ^  =  ^11^  ioTt. 

8.  Solve  !^^±I--l_  =  ??'i:^  forr. 
n-\-r      ir  —  'tr      n  —  r 

9.  Solve  i4--  +  i  =  lforP. 

p    q    R 

10.  Solve  -i  + 1  -  ~  +  — ^=  0  for  d. 

N  d      dN 

11.  Solve  ^±^  +    ,  y       =    /        for  *. 

12.  Solve  ^-AtiO_4_^+3^^J_f,,^^ 

3  ^  4  ^  10 

13.  Solve  2Z+2Z P+TT         ^  P+ TT  f^,  p. 

14.  Solve  ^f +  f  =  -^  4- 1  for  c. 

c^  —  c^^       c  +  d 

Note.  —  The  following  formulse  express  important  principles  in  science, 
geometry,  etc.  It  often  is  necessary,  in  using  such  formulse,  to  express  one 
of  the  quantities  involved  in  terms  of  the  others.  The  pupil  will  find  it  of 
decided  advantage  to  learn  that  process  here. 

15.  If  O  and  c  are  the  circumferences,  and  E  and  r  the  radii, 
respectively,  of  any  two  circles,  then 

or 

Solve  for  r. 


FRACTIONAL   EQUATIONS 


199 


16.  The  figure  bounded  by  two  radii  of  a  circle  and  the  arc 
subtended  by  them  is  a  sector  of  the  circle.  If  s  and  a  are  the 
area  and  arc,  respectively,  of  a  sector  of  a  circle  whose  area  is  S 
and  circumference  (7,  then       £.  _  ^ 

Solve  for  s ;  iov  S)  f or  a ;  for  C, 

17.  If  one  gear  or  cogwheel  drives  another,  and  S  and  T  are 
the  speed  and  number  of  teeth  of  the  driving  wheel,  and  s  and  t 
the  speed  and  number  of  teeth  of  the  driven  wheel,  respectively, 

then  —  =  — 

s      T 

Solve  for  S ;  for  s ;  f or  « ;  for  T. 

If  the  number  of  teeth  of  the 
driving  wheel  is  96,  of  the  driven 
wheel  12,  and  the  speed  of  the 
driving  wheel  40  revolutions  a 
minute,  find  the  speed  of  the  driven  wheel. 

18.  The  formula  for  computing  the  area  iv  of  a  lune,  the  por- 
tion of  the  surface  of  a  sphere  bounded  by  two  semicircles  such 
as  the  meridians  on  the  earth's  surface,  is 

ST      4  ' 
Solve  for  L ;  for  A.  "^ 

19.  In  two  similar  right  cylinders  whose  volumes,  altitudes, 
and  radii  are  F,  H,  R,  and  v,  hy  r,  respectively, 

Solve  for  F;  for  y;  for  H;  for  h. 

20.  If  T  and  t  are  the  areas  of    

the   entire   surfaces   of    the   above 
cylinders,  t  ^  R(H+E) 

t        r(Ji-\-r) 
Solve  for  T;  fort;  for  H]  for  h. 


800  ELEMENTARY  ALGEBRA 

21.  If  the  radius  of  a  right  cylinder  is  denoted  by  i?,  the  area 
of  the  curved  surface  by  S,  the  area  of  the  total  surface  by  T,  the 
altitude  by  -ET,  and  the  volume  by  V, 

(1)  It  =  ^;        (2)  RH^f;        (3)  U^H^E; 

(4)  R'  +  HR  =  ^',        (5)  R'-\-HR  =  ?^^ 

Solve  (1)  for  F;  for  S.  Solve  (2)  for  i?;  for  H;  for  >S'.  Solve 
(3)  for  H'y  for  F.  Solve  (4)  for  H;  for  T.  Solve  (5)  for  T;  for 
/S;  for^. 

22.  In  the  above  cylinder  it  is  known  also  that  —  =  — - — • 
Solve  for  H;  for  R;  for  T;  for  S.  MS 

23.  The  formula  F=  — ^  is  used   in  computing  centrifugal 

r 

force,  or  the  force  with  which  an  object  moving  in  a  circle  tends 
to  fly  off  from  the  center.     It  is  the  force  which  causes  grind- 
stones and  flywheels  to  burst  when  run  at  too  high  a  speed,  and 
carriages  to  overturn  when  turning  a  corner  at  too  high  a  speed. 
Solve  for  M)  for  r. 

24.  Solve  /=  —  for  w :  for  g :  for  r. 

gr 


25.  The  densities  of  some  substances  are  computed  by  the 
cmula, 

Solve  for  w\  for  w\ 


formula,  ^ 


26.   The  formula  (7  =  —  is  of  great  importance  in  electricity. 
R 

It  gives  the  relation  between  the  quantity  of  current  C  of  elec- 
tricity flowing  through  a  wire  or  other  conductor,  the  resistance 
offered  to  the  current   by  the  conductor,  and  the  electrical  pres- 
sure (electromotive  force)  required  to  overcome  that  resistance. 
Solve  for  ^;  ioi  R, 


FRACTIONAL  EQUATIONS 


201 


27.  If  the  resistance  within  an  electric  battery  cell  is  R,  the 
resistance  of  the  external  circuit  through  which  the  current  flows 
r,  the  strength  of  current  G,  and  the  electric  pressure  E,  then 

E  +  r' 
Solve  for  E ;  for  E ;  for  r. 

28.  If  n  equal  electric  cells  are  connected 
by  wires  in  one  way  (series),  so  as  to  form  a 
battery,  the  current  strength  is  found  by  the 
formula 


nE 


R-{-nr 
Solve  for  n ;  for  E ;  for  R ;  for  r. 

29.   If  the  cells  in  Problem  28  are  connected  in  another  way 
(parallel),  the  current  strength  of  the  battery  is  found  by  the 


formula 


C: 


E 


rV- 

n 


Solve  for  n ;  for  E)  for  R\  for  r. 

30.  If  an.  electric  circuit  between  two  jjoints  is  divided  into 
two  branches,  so  that  part  of  the  current  flows  through  one  branch 
and  the  rest  through  the  other,  the  total  resistance  of  the  circuit 
is  computed  by  the  formula 

R      Vi     r^ 

where  R  is  the  total  resistance,  ?'i  the  resistance  of  one  branch, 
and  Vz  the  resistance  of  the  other  branch. 
Solve  for  R. 

31.  If  the  circuit  in  Problem  30  is  divided  into  three  branches 
whose  resistances  are  ri,  o^,  and  r^,  the  formula  is 

R     ri     r,     ra 

Solve  for  R, 


202  ELEMENTARY  ALGEBRA 

32.  Solve /S  =  ^^=^^  for  r;  for  a;  f or  ^. 

33.  The  radius  r  of  the  arch  of  a  bridge  or  doorway  whose  span 

is  w  and  rise  h  is  found  from  the  formula 

Solve  for  r. 

34.  Find  the  radius  at  which  the  stones  must  be  cut  for  an 
arch  whose  span  is  16  ft.  and  rise  4  ft. 


Solve 


SUPPLEMENTARY  EXERCISES 

1.  „-.^-^+       ^  ^ 


2n^-n-l     2w2  4-3n  +  l      w^-l 


2.        1 


^-2      ^+2      ^-3 

7^  +  2 w  +  2         ^       2 

6/2-1  3r-5  .  o 


r2-9r  +  14       r-7 

18^-27      11^-1^9^  +  11. 
14  3^  +  1  7 

Suggestion.  —  In  equations  like  this,  in  which  some  denominators  are 
monomials  and  others  polynomials,  it  often  is  best  to  clear  of  monomial 
, denominators  only  at  first,  and  then  simplify  before  clearing  of  the  remaining 
denominators.     Thus,  multiplying  both  members  by  14, 

18  J  -  27  +  IMliril  =  18  «  +  22. 


This  becomes  IhU-U  ^  ^g^ 


U-1 

«  +  l 


FRACTIONAL   EQUATIONS  203 

4P4-3      7P-29^8P4-19 
9  5P-12  18 

5Jc  +  5^nic-15     33A;  +  15. 

*  k-5  10  30 

8    ±\      -^  +  ^         9iVr+3^3i^+6 

*  2(3iV^-4)"^      15  5 

g     a?  — '^       a;  — 9_ar  — 13      x  —  15 
'   x  —  9      ic— ll~a;  — 15      ic  — 17* 

Suggestion.  —  In  equations  of  this  kind  much  multiplication  may  be 
avoided  by  combining  some  of  the  fractions  before  clearing  the  equation 
of  fractions.     Thus,  performing  the  subtractions  indicated  in  each  member, 


a;2  -  20  a;  +  99      x^  -  32  a;  +  266 

10.    y-5  ^  y-7^y-4  ^  y-   8 
y  — 7     y  —  9     y-6     y  —  10 

Suggestion.  — First  transpose  iu  order  to  form  differences  such  that  the 
fractions  when  combined  will  give  simple  numerators.     Thus, 

y-5     y— 4__y-   8     y  — 7 
y_7      y_6     y-10     y-9* 

11        J3        B-tl^B-S     .8-9. 
•    B-2     B-1     B-6     B-f 

1111  r 


12. 


V— 3     v—1     v— 4     V— 2 


,o     w  — 3     n  — 4     n  — 1  ,  n  — 2 

18. -  = --\-- • 

«— 4     w— 5     n—2     6—n 


14     <»-M      a  +  6^a4-5     a-f-2^ 
a  +  2     a  +  7      a  +  6     a-j-S' 


204  ELEMENTARY  ALGEBRA 


jg^    P-1  .  P-7      P-n     P-3 


P-2     P-8     P-6     P-4 


16      ^^     A;-5_A;-10      >[;-4 

^5"^^9 ~ Xf4  "^^+10* 


17  2aH-5  2a  +  2^2a4-l  2aH-6 
2a  +  6  2a-i-3  2aH-2  2a  +  7* 


18.  ?:i^=_L_+ri:i?. 

r-2     r2-r-2^r  +  l 


19    4y  +  l_   4y-l^        8 
42/-1     4y  +  l      16/-1' 


_z 3-2g^  3(2 g~ 7)^ 

3-«      3  +  ;3        6(3  +  »)' 


21.   i^  =  -^+     3 


s  — 2     S4-2     s  +  1 


22.   Solve  ^^±^~   ^^-^   =^'  +  <^  +  ^)fora; 
5a;H-c       6a;  +  2c         6(6a;  +  3c) 


28.   Solve  - — ^^^ +  £ll?  =  0  for  I. 

V  +  ct  —  at  —  ao     t  —  e 


24.   Solve -f — ^ =  — ^ forn. 

a(p  —  n)      b(c  —  n)      a(c  —  n) 

«..   solve  f-^(|-|)=^(„-?M)fo.. 


FRACTIONAL  EQUATIONS  205 


26.   Solve  i  =  i+i+i  +  i  fori?. 

B    Tx    n    n    n 


27.   Solve  E  = ^ — j-r-  for  <, 


^-m 


28.   Solve  ^^V^ +  1=,   .     ^    ,forr. 

m  — w 


29.  In  a  guessing  game,  the  leader  says :  "  If  you  will  add  20 
years  to  your  age,  divide  the  sum  by  your  age,  add  3  to  the  quo- 
tient, and  tell  me  the  result,  I  will  tell  you  your  age."  How  did 
he  find  it  ? 

Suggestion.  — Let  x  =  the  age  and  a  =  the  result  given.  Write  an  equa- 
tion and  solve  it  for  x. 

30.  Make  up  a  guessing  game  similar  to  that  in  Problem  29, 
and  show  how  the  age  is  found. 

31.  If  a  body  whose  weight  is  W  pounds,  moving  with  a  ve- 
locity of  Ffeet  per  second,  strike  a  second  body  whose  weight  is 
w  pounds  and  which  is  at  rest,  so  that  the  two  bodies  move  on 
together,  the  velocity  v  with  which  they  continue  together  is 
found  from  the  formula 

W+w 
Solve  for  W;  for  F;  for  w, 

32.  If  a  freight  car,  weighing  40,000  lb.  and  moving  8  ft.  per 
second,  strike  a  second  car  that  weighs  32,000  lb.  and  is  standing 
still,  and  the  two  are  then  coupled  together,  with  what  speed  will 
they  continue? 


CHAPTER   XI 
PROPORTION.    VARIABLES 

114.  Ratio.  —  The  quotient  of  one  number  divided  by  another 
of  the  same  kind  is  sometimes  called  their  ratio.  It  usually  is 
written  in  the  form  of  a  fraction,  and  is  subject  to  all  of  the 
rules  that  apply  to  fractions. 

Thus,  the  ratio  of  .$  6  to  $  100  is  written  -^^.     The  ratio  of  2  lb.  to  5  lb. 

2  1b  ^^^^ 

is  written  '-,    And,  in  general,  the  ratio  of  a  to  5  is  written -. 

5  lb.  b 

Note.  —  An  old  form  of  writing  the  ratio  of  a  to  &  is  a  :  &.  This  notation 
is  less  convenient  for  computation  than  the  fractional  form,  and  is  used  less 
now  than  formerly. 

The  dividend  or  numerator  of  a  ratio  is  sometimes  called  the 
antecedent,  and  the  divisor  or  denominator  is  called  the  consequent. 

EXERCISES 

1.  Express  the  ratio  of  3  qt.  to  6  qt.  as  a  fraction  in  its  lowest 
terms. 

2.  If  the  rate  of  interest  on  a  sum  of  money  is  5  %,  that  is, 
the  ratio  of  $5  to  $100,  express  the  rate  as  a  fraction  in  its 
lowest  terms. 

3.  The  death  rate  in  Boston  in  a  recent  year  was  18  to  1000 
population.     Express  this  ratio  as  a  fraction. 

4.  An  alloy  consists  of  copper  and  tin  in  the  ratio  of  2  to  3. 
What  part  of  it  is  each  ? 

5.  A  solution  consists  of  alcohol  and  water  in  the  ratio  of  3 
to  5.     What  part  of  it  is  water  ? 

206 


PROPORTION.     VARIABLES 


207 


6.  In  a  city  of  84,000  population  the  number  of  births  in  a 
year  is  560.     What  is  the  birth  rate  ?     How  many  per  1000  ? 

7.  When  a  man  5  ft.  10  in.  tall  casts  a  shadow  35  ft.  long, 
find  the  ratio  of  his  height  to  the  length  of  his  shadow. 

8.  The  specific  gravity  of  a  solid  or  liquid  is  the  ratio  of  the 
weight  of  any  volume  of  it  to  the  weight  of  an  equal  volume  of 
water.  A  cubic  foot  of  water  weighs  62.5  lb.  A  cubic  foot  of 
steel  weighs  490  lb.  Find  the  specific  gravity  of  steel.  Express 
the  answer  first  as  a  common  fraction,  then  as  a  decimal  computed 
to  tenths. 

9.  Find  the  specific  gravity,  computed  decimally  to  tenths,  of 
each  of  the  following  substances : 


Substance 

Weight  in  PorNDS 
OF  1  Cr.  Ft, 

SlBSTANCE 

Weight  in  Pounds 
OF  1  Cu.  Ft. 

Cast  Iron 

Brass 

Gold 

450.0 

523.8 

1200.9 

Oak 
Ash 

Cork 

65.0 

52.8 
15.0 

10.  The  diameter  of  a  circular  plate  is  6  in.,  and  the  circum- 
ference is  measured  with  a  tape  line  and  found  to  be  18.85  in. 
Find  the  ratio  of  the  circumference  to  the  diameter. 

11.  A  boy  measures  the  wheel  of  his  bicycle,  and  finds  that  its 
diameter  is  28  in.  and  circumference  88  in.  What  is  the  ratio  of 
the  circumference  to  the  diameter  ? 

12.  Simplify  the  ratio  of  x-\-y  to  3?^y^  by  writing  it  as  a 
fraction  reduced  to  its  lowest  terms. 

13.  Simplify  the  ratio  of  -  to  —  by  writing  it  as  a  simple 
fraction.  ^ 

14.  Which  ratio  is  the  greater,  -^  or  |f  ? 

Suggestion.  — Reduce  thera  to  a  common  denominator,  and  compare  the 
numerators. 

15.  Write  in  descending  order  of  magnitude  |,  |f,  |^. 


208  ELEMENTARY  ALGEBRA 

16.  Two  partners  in  business,  A  and  B,  divide  a  profit  of 
$  2135  between  them  so  that  A's  part  and  B's  part  are  in  the  ratio 
of  3  to  4.     How  many  dollars  does  each  receive  ? 

Suggestion.  —  Let  d  be  the  amount  that  A  receives. 
Then,  2135  —  (Z  is  the  amount  that  B  receives. 

Hence,  ^ =  -. 

'  2135 -(^     4 

17.  If  two  partners,  A  and  B,  divide  a  profit  of  $1200  in  the 
ratio  of  1  to  2,  how  many  dollars  does  each  receive  ? 

18.  In  a  certain  city  with  a  population  of  247,520,  the  ratio  of 
the  Germans  to  all  other  nationalities  together  is  2  to  15.  What 
is  the  German  population? 

19.  Separate  40  into  two  parts  which  are  in  the  ratio  of  3 
to  7. 

20.  Separate  85  into  two  parts  which  are  in  the  ratio  of  5 
to  12. 

21.  The  sides  of  a  triangle  are  5  in.,  9  in.,  and  7  in.  Divide 
the  side  7  in.  long  into  two  parts  whose  ratio  equals  the  ratio  of 
the  other  two  sides. 

115.  Proportion.  —  An  equation  whose  members  consist  of  two 
ratios  is  called  a  proportion. 

Thus,  f  =  |,    ^  =  Jl^,  and  f  =  ^  are  proportions. 
486oloo  0      d 

The  proportion  -  =  - ,  is  read  either  "  a  over  b  equals  c  over  d," 

0        Ct/ 

or  "  a  is  to  6  as  c  is  to  c?." 

The  numbers  forming  one  of  the  ratios  are  said  to  be  "propor- 
tional to"  the  numbers  forming  the  other  ratio;  or  the  four  num- 
bers forming  the  ratios  are  said  to  be  "in  proportion." 

•  Ob  C 

In  any  proportion  -  =  - ,  a,  6,  c,  and  d  are  called  the  terms.    The 

first  and  fourth  terms,  a  and  d,  are  called  the  extremes ;  and  the 
second  and  third  terms,  h  and  c,  are  called  the  means. 

Since  a  proportion  is  an  equation,  all  of  the  operations  which 


PROPORTION.     VARIABLES  209 

may  be  performed  upon  an  equation,  such  as  clearing  of  fractions, 
etc.,  may  be  performed  upon  a  proportion. 

Note.  — Since  the  ratio  of  any  two  concrete  numbers  is  the  same  as  that 
of  the  corresponding  abstract  numbers,  in  performing  any  operation  upon  a 
proportion,  confusion  ma^  be  avoided  by  using  only  abstract  values  for  the 
terms. 

EXERCISES 

Find  the  value  of  the  unknown  term  in  each  of  the  following 
proportions : 


,     n     15 

.    7     42 

36_1 

1-    x  =  Tr' 

4.    -  = 

7. 

2      10 

9       t 

V      4' 

a;      1 

3  _^ 

9  _12 

8. 

*   12     8* 

*  10     40* 

16~P 

«    4     16 

24      r 

20      8 

3.   —  =  — -• 

9. 

— .  —  —  • 

y    21 

•  35      70* 

36     L 

Find  the  values  of  n  in 

the  proportions : 

10.  1  =  '-. 

12.  i=». 

14. 

l-L. 

0      n 

V        W 

r      n 

11.2  =  2. 

13.    »-«  =  « 

+  ^. 

15. 

an  _h 

n      z 

a 

h 

h  ~c' 

16.  A  family  of  three  members  and  a  family  of  four  members 
camp  out  together.  The  total  cost  of  provisions  is  $  112,  which 
they  wish  to  divide  in  proportion  to  the  sizes  of  the  families. 
How  much  must  each  family  pay  ? 

116.    Definitions.  —  In  any  proportion  -  =  -,  the  fourth  term  d 

b      d 
is  called  a  fourth  proportional  to  the  other  three  terms,  a,  b,  and  c. 

A  proportion  such  as  -  =  - ,  in  which  the  means  are  equal,  is 

called  a  mean  proportion.     The  mean  term  x  is  called  the  mean 
proportional  between  the  extremes  a  and  b. 

In  a  mean  proportion  such  as  -  =  -,6  is  called  the  third  pro» 
portional  to  a  and  x.  ^     o 


210  ELEMENTARY  ALGEBRA 

EXERCISES 

Find  the  fourth  proportional  to ; 

1.  2,  3,  and  10. 

Suggestion.  —Let  x  be  the  fourth  proportional.    Then  -  =  — . 

3       X 

2.  4,  5,  and  12.  5.   1,  a,  and  a?, 

3.  7,  10,  and  28.  6.   m,  n,  and  mH, 

4.  16,  12,  and  4.  7.   1.  1  -  t,  and  1  +  ^. 

Find  the  third  proportional  to : 

8.  3  and  6. 

Suggestion.  —  Let  x  be  the  third  proportional.     Then  -  =  -. 

6      X 

9.  8  and  12.  12.    a  and  h, 

10.  ajand3^.  13.    1  +  ^  and  1  —  w^. 

11.  Pandit.  14.   10  and  1. 

Find  the  mean  proportional  between : 

15.  2  and  8. 

Solution.  —  Let  x  be  the  mean  proportional.     Then  =  =  5. 
Clearing  of  fractions,  ic^  =  16. 

Taking  square  root,  a;  =  i  4. 

16.  3  and  12.  19.    n  and  n\ 

17.  7  and  28.  20.   2  A"^  and  50  A 

18.  9  and  1.  21.    i^  and  i2  +  2  i?^  +  R\ 

117.   Distances   found  by  Proportion.  —  Two  triangles  that  are 
drawn  with  the  angles  of  one  equal  to  the  corresponding  angles  of 

the    other    have    the    same 
shape,  and  are  called  similar 
triangles,  as  ABC  and  DEF, 
^\.                  X                \^         Let   the   student  draw   two 
A-^^ ^B       D^^ ^F     such   triangles,  making   the 


PROPORTION.     VARIABLES  211 

side  DE  of  one  three  times  as  long  as  the  corresponding  side  AB 
of  the  other.  Measure  the  other  sides,  and  compare  them.  It 
will  be  found  that: 

In  two  similar  triangles,  the  corresponding  sides  are  proportional. 

AB      AC 
Thus,  in  the  above  figure,  —  =  — - ,  etc. 

This  principle  is  much  used  in  computing  unknown  distances. 


EXERCISES 

1.  In  two  similar  triangles,  the  sides  of  the  larger  are  12  in., 
14  in.,  and  20  in.,  and  the  shortest  side  of  the  smaller  is  3  in. 
Find  the  other  sides  of  the  smaller  triangle. 

2.  The  sides  of  a  triangular  field  are  16  rd.,  24  rd.,  and  30  rd. 
The  smallest  side  of  a  similar  field  is  40  rd.  Find  the  other  two 
sides. 

3.  A  tree  casts  a  shadow  48  ft.  long  when  a  vertical 
rod  6  ft.  high  casts  a  shadow  4  ft.  long. 
How  high  is  the  tree  ?  /I  /  ^^j^ 

4.  A  church  spire  casts  a  shadow  23     _/^J-^-'^ 
ft.  long  when  a  man  5  ft.  10  in.  tall  -j^--  - 
who  is  passing  by  casts  a  shadow  2  ft.  long.     Find  the  height  of 
the  spire. 

5.  The  distance  from  ^  to  ^  on  opposite  sides  of  a  lake  may 

^^^^^  be  found  as  follows : 

A 's^j^^^^^p^B       The  distances  from  A  to  R  and  from  B 
^v^^,.^^^         to  T  are  measured  off,  making  the  triangles 
^.^.^"^X^^  ASB  and   TSR  similar.     If  AS  is   taken 

'^^=^— -^R        400  yd.,  and  SR  300  yd.,  and  TR  is  meas- 
ured and  found  to  be  580  yd.,  how  far  is  it  from  ^  to  J5  ? 

6.  If  A  and  B  are  two  points  on  opposite  sides  of  a  hill,  and 
out  in  the  plain  at  the  foot  of  the  hill  distances  are  measured  as 
in  Problem  5,  so  that  AS  is  taken  4050  ft.  and  SR  1350  ft.,  and 
TR  is  found  to  be  1600  ft.,  how  far  is  it  between  A  and  B  ? 


212 


ELEMENTARY  ALGEBRA 


7.  To  find  the  distance  AB  across  a  stream,  measure  off  a  dis- 
tance -4(7  several  yards  long,  along  the  bank.     Then  at  C  measure 

off  a  distance, (7Z>  at  right  angles  to  AC.  By 
sighting  across  from  D  to  J5,  locate  a  stake  at  a 
point  E  of  AC  in  line  with  D  and  B.  Get  the 
lengths  of  AE  and  EC.  Since  the  triangles  are 
similar,  write  the  proportion  by  means  of  which 
AB  may  be  computed. 

If  AE  is  80  yd.,  EC  20  yd.,  and  DC  15  yd., 
what  is  the  width  of  the  stream  ? 

8.  It  is  known  in  geometry  that  a  line  parallel  to  one  side  of 
a  triangle  divides  the  other  two  sides  into  four  proportional  parts. 

That  is,  if  DE  is  parallel  to  AB,—  =  —  .  ^ 

^  '  DA     EB  ^ 

(1)  li  AD  =  6  in.,  DC  =  3  in.,  and  BE  y^\ 
=  4  in.,  find  EC.                                                          PX^  \e 

(2)  If  AD  =  10  in.,  BE  =12  in.,  and  EC    ^^  Xp 
=  20  in.,  find  DC 

(3)  If  DC  =  9  in.,  BE  =T  in.,  and  EC  =  5  in.,  find  AD. 

9.  In  the  figure  of  Problem  8,  if  AD  =  S  in.,  DC  =5  in.,  and 
BC=26  in.,  find  BE  and  EC. 


Suggestion. 
Hence, 


Let  BE 
8. 
6 


X.     Then  EG  =  26-  x. 

X 


26- a; 

10.  In  the  figure  of   Problem  8,  if  AC  =  4.0  ft.,  J5^^=12  ft., 
and  EC  =15  ft.,  find  AD  and  DC. 

11.  It  is  known  in  geometry  that  in  any  triangle  ABC,  if  the 
line  CD  divides  the  angle  at  C  into  two  equal  parts,  it  divides 

Q  the  opposite  side  into  parts  proportional  to  the 

AD 


other  two  sides ;  that  is,  = 


AC 
DB      BC' 

If  AC=  15  in.,  BC  =  10  in.,  and  AB  =  12  in., 
find  AD  and  DB. 

12.    In  the  figure  of  Problem  11,  if  AC  =  AO  yd.,  BG=  32  yd., 
and  AB  =  60  yd.,  find  AD  and  DB. 


B 


PROPORTION.     VARIABLES 


213 


13.    In  the  triangle  ABC,  the  angle  at  0  is  a  right  angle,  and 
CD  is  perpendicular  to  AB.    It  is  known 
ill  geometry  that  CD  is  a  mean  propor- 
tional between  AD  and  DB. 


If  AD  =20 
CD. 


in.,  and  DB  =  5  in.,  find 


14.  A  method  used  several  hundred  years  ago  of  finding  the  dis- 
tance from  A  to  the  inaccessible  point  B  was  to  erect  a  vertical  staff 

ACj  place  upon  it  an  instrument  resembling  a 
carpenter's  square,  pointing  one  blade  towards 
B,  and  note  the  place  D  on  the  ground  to  which 
the  other  blade  pointed.  ACsnid  DA  were  meas- 
ured.   If  AC  =52  in.,  and  DA=6  in.,  find  AB. 

15.  In  the  semicircle  with  the  diameter  AB,  CD,  a  perpendic- 
ular to  AB,  is  a  mean  proportional  between  AD  and  DB. 

If  AD  =  12  in.,  and  DB  =  3  in.,  find  CD.  ^ X 

If  ^D  =  8  in.,  and  CD  =  6  in.,  find  DB. 
If  AB  =  20  in.,  and  CZ>=8  in.,  find  AD  and 
DB. 

118.  Proportion  in  Simple  Machines.  —  Simple  machines,  such  as 
the  lever,  the  wheel  and  axle,  etc.,  are  instruments  by  means  of 
which  a  small  effort  exerted  may  be  made  to  overcome  a  resistance 
of  much  greater  size.  Proportion  is  involved  in  the  use  of 
simple  machines. 


EXERCISES 

1.  A  lever  is  a  stiff  bar  of  wood  or  metal  that  is  movable 
about  a  fixed  point  or  pivot  called  the  fulcrum.  The  resistance 
W  to  be  overcome  is  applied  at  one  end  of  the  bar,  and  the  effort 
w,  exerted  to  overcome  it,  at  the  other  end.  If  D  is  the  dis- 
tance of  W  from  the  fulcrum,  and  d  the  distance  of  vo  from  the 
fulcrum,  then 


214 


ELEMENTARY  ALGEBRA 


What  effort  is  required  to  lift  a  weight  of  600  lb.  by  a  lever, 
if  the  weight  is  8  in.  from  the  fulcrum  and  the  effort  applied  40 
in.  from  the  fulcrum  ? 

2.  What  effort  is  required  to  lift  a  stone  weighing  840  lb.  by 
means  of  a  crowbar  60  in.  long,  if  the  fulcrum  is  placed  4  in.  from 
the  stone  ? 


3. 


A  blacksmith  weighing  180  lb.  lifts  wagons,  etc.,  by  means 
of  a  wagon  jack.  The  weight  lifted  is  applied 
at  a  point  6  in.  from  the  pivot  or  fulcrum,  and 
he  grasps  the  handle  32  in.  from  the  pivot.  By 
throwing  his  whole  weight  upon  the  jack, 
how  much  can  he  lift  ? 

4.  The  entire  length  of  a  lever  is  48  in.  Where  must  the 
fulcrum  be  placed  in  order  that  a  resistance  of  360  lb.  may  be 
overcome  by  an  effort  of  120  lb.  ? 

5.  The  wheel  and  axle  consists  of  a  wheel  and  a  cylindrical 
axle  passing  through  its  center,  the  two  being  fastened  rigidly 

together,  so  that  the  axle  is  turned  by  revolving 
the  wheel.  The  weight  or  resistance  to  be  over- 
come is  applied  at  tlie  circumference  of  the  axle 
through  a  cord  wrapped  around  the  axle,  and 
the  effort  required  to  overcome  it  is  applied  at 
the  circumference  of  the  wheel.  If  the  resist- 
ance is  W,  the  effort  required  to  overcome  it  w, 
the  radius  of  the  axle  B,  and  the  radius  of  the 

wheel  r,  then  —  =  — . 
W      r 

If  the  radius  of  the  wheel  is  20  in.,  the  radius  of  the  axle  6  in., 

and  the  resistance  to  be  overcome  500  lb.,  what  is  the  effort 

required  ? 

6.  The  axle  of  a  windlass  shown  in  the  above  figure,  which  is 
used  for  drawing  water  from  a  well,  has  a  radius  of  4i  in.,  and 
the  radius  of  the  wheel  15  in.  What  effort  is  required  to  lift  a 
bucket  of  water  weighing  54  lb.  ? 


PROPORTION.     VARIABLES  216 

7.  A  capstan,  a  form  of  wheel  and  axle  used  in  lifting  anchors 
on  ships,  has  an  axle  12  in.  in  diameter  and  a  lever  arm  or  spoke 
45  in.  long.  How  much  effort  is  required  to  lift  by  it  an  anchor 
weighing  a  ton  ? 

8.  The  inclined  plane  is  a  smooth  sloping  surface  that  is  used 
in  raising  a  heavy  object  through  the  application  of  a  compara- 
tively small  effort,  the  object  being  raised  by  sliding  or  rolling  it 
along  the  surface  of  the  plane.  If  the  effort  is  applied  parallel  to 
the  surface  of  the  plane,  the  effort  applied  is  to  the  weight  lifted 

as  the  height  of  the  plane  is  to  its  length ;  that  is,  -^=  — ,  where 

w  =  the  effort  applied,  W  =  the 
weight,  H  =  the  height  of  the 
plane,  or  distance  the  weight  is 
lifted,  and  L  =  the  length  of  the 
plane. 

If  the  length  of  an  inclined 
plane  is  12  ft.,  the  height  3  ft., 
and  the  weight  of  the  object  lifted 
400  lb.,  what  effort  is  required  to  move  it  up  the  plane  ? 

9.  A  barrel  of  material  weighing  180  lb.  is  loaded  into  a  wagon 
bed  32  in.  from  the  ground  by  rolling  it  up  a  board  10  ft.  long 
with  one  end  resting  on  the  ground  and  the  other  on  the  wagon 
bed.     What  effort  is  required  ? 

10.  Two  men  place  a  580  lb.  building  stone  upon  a  wagon  by 
sliding  it  up  a  board  8  ft.  long,  one  end  of  the  board  being  on  the 
ground  and  the  other  resting  on  the  bed  of  the  wagon.  The  wagon 
bed  is  20  in.  from  the  ground.  What  effort  do  they  use  in  loading 
the  stone  in  addition  to  that  required  to  overcome  the  friction  ? 

11.  Ice  is  stored  in  an  ice  house  by  dragging  it  up  an  inclined 
plane.  The  incline  is  200  ft.  long  and  40  ft.  high.  What  effort 
is  required  to  pull  up  a  block  of  ice  weighing  250  lb.  ? 

12.  A  locomotive  pulls  a  train  weighing  720  tons  up  a  grade 
with  a  rise  of  2^  ft.  to  100  ft.  of  grade.  How  much  more  pull  must 
the  locomotive  exert  than  if  the  train  were  on  a  level  road  bed  ? 


216  ELEMENTARY  ALGEBRA 

119.   Important  Principles  in  Proportion.  —  The   following  im 
portant  principles  in  proportion  are  often  used  in  geometry,  and 
elsewhere,  where  proportion  is  applied.     Let  the  pupil  establish 
each  of  them. 

(1)  If  four  numbers  are  in  proportion^  the  product  of  the  extremes 
equMs  the  product  of  the  means.     That  is, 

if  ^  =  £  then  ad  =  be. 


Suggestion.  —  Clear  -  =  -  of  fractions. 
b      d 

(2)  If  the  product  of  two  numbers  equals  the  product  of  two  other 
numbers  J  the  four  numbers  are  in  proportion.     That  is, 

if  ad  =  be,  then  -  =  -  • 
'  b      d 

Suggestion.  —  Divide  both  members  of  ad  =  he  by  hd. 

(3)  If  four  numbers  are  in  proportion,  they  are  in  proportion  by 
inversion.     That  is, 

if  2  =  «,  then5  =  ^. 
b      d  a     G 

Suggestion.  —  Divide  1  =  1  by  the  members  of  -  =  - . 

b      d 

(4)  In  any  proportion^  the  means  may  be  interchanged,  or  the 
extremes  interchanged,  without  destroying  the  propmi,ion.     That  is, 

if  »  =  £,  then«  =  *  and^  =  ^. 
b      d  c      d  b      a 

Suggestion.  —  Multiply  both  terms  of  -  =  -  by  - ;  by  - . 

b      d       c  a 

(5)  Tlie  terms  of  any  proportion  are  in  proportion  by  addition. 
That  is, 

if^  =  ^,  then^±^  =  £±^. 
b      d'  b  d 

Suggestion.  —  Add  1  to  each  member  of  -  =  -  • 

b      a 


PROPORTION.    VARIABLES  217 

(6)    The  terms  of  any  proportion  are  in  proportion  by  subtraction. 
That  is, 


b      d'  b  d 

b     d 


d      c 

Suggestion.  —  Subtract  1  from  each  member  of 


(7)  Like  powers  of  the  terms  of  a  proportion  are  in  proportion. 

That  is, 

if  «  =  ^,  then^  =  |,   ±'  =  4  etc. 

Suggestion.  —  Raise  both  members  of  -  =  -  to  the  same  power. 

h     d 

(8)  If  two  or  more  ratios  are  equal,  the  sum  of  the  antecedents  is 
to  the  sum  of  the  consequents  as  any  antecedent  is  to  its  consequent. 
That  is, 

•^  a     c      e         .       .■,        aH-c  +  e4-  etc.      a        . 

if  -  =  -  =  —  =  etc.,  then     ^    ^    ^ =  -=  etc. 

b      d     f  b-\-d+f+  etc.      b 

For,  since  -  =  -  =  -  =  etc.,  let  each  ratio  equal  r. 
b     d     f 

Then  a  =  rb,  c  =  rd,  e  =  rf,  etc. 

Hence,  a  +  c-h  e  +  etc.  =  rb  ■}- rd  -\- rf  +  etc. 

=  r(6  +  <Z+/+etc.). 

Hence,  «i^Ji.^-  +  ^  =  r  =  «  =  ^  =  etc. 

b  +  d+f+etc.  b     d 


EXERCISES 


1.   If  ad  =  be,  show  that  -  =  —  < 
c      d 


2.  If  ad  =  6c,  show  that  ^  =  -. 

b     a 

3.  If  ad  =  be,  show  that  -  ==  - . 

c      a 


218  ELEMENTARY  ALGEBRA 

4.    If  -  =  - ,  snow  that  — - —  =  — ' 

b      d  a—bc—d 

Suggestion.  — Divide  the  members  of  .^  "^     =  ^-i—  (5)  by  the  members 
1,  7  b  a 

-^^a-b^c-d  .Q. 

b  d      ^  ^ 

,      5.    If  ^  =  ^,  show  that  ^^+^  =  ^. 
b      d  .  c+d     d 

6.  If  -  =  -,  show  that     ~    =  - • 

b      d  c—dd 

7.  If  ^  =  '-,  show  that  i^^±*  =  ^«±^. 

b      d  b  d 

8.  If  «  =  £,  show  that  ^^1^  =  ^:^^. 

b     d'  b  d 

9.  It  is  known  that,  in  triangle  ABC,  if  DE  is  parallel  to  AB, 

AD     BE      at,      ^        4.1,  •   4.1.  i.  -^O     BC 
_  =  — .     Showfromthisthat— =  — . 

10.   In  the  similar  triangles  in  §  117,  it  is 
^  \p  known  that  ^  =  ^=  ^.      Show  from 

this  that  the  ratio  of  the  perimeters  of  the  triangles  equals  the 
ratio  of  any  two  corresponding  sides,  such  as  AB  and  DE.  See 
§  119,  (8). 

120.  Variable  Quantities.  —  Any  quantity  such  as  one's  age,  the 
temperature,  the  population  of  the  country,  etc.,  that  is  always 
changing  is  called  a  variable  quantity,  or  a  variable.  Any  quantity 
that  does  not  change  is  called  a  constant. 

Many  of  the  quantities  encountered  in  business,  in  science,  etc., 
are  variables. 

Thus,  the  prices  at  which  many  foodstuffs  are  retailed  usually  vary  from 
day  to  day,  or  from  week  to  week.  Butter  may  sell  at  24  ^  one  week,  27  ^ 
the  next  week,  29^  the  next  week,  etc. 

As  an  illustration  of  variable  quantities  encountered  in  science,  suppose 
that  a  small  heavy  object  such  as  a  lead  ball  be  dropped  from  a  great  height 
toward  the  earth.  The  farther  it  falls,  the  faster  it  falls.  At  the  end  of  one 
second  its  speed  will  be  about  32  ft.  per  second,  at  the  end  of  the  second 


PROPORTION.     VARIABLES  219 

second  its  speed  will  be  64  ft.  per  second,  at  the  end  of  the  third  second  its 
speed  will  be  96  ft.  per  second,  etc.  That  is,  the  speed  is  a  variable  con- 
stantly increasing. 

121.  Related  Variables.  —  Some  variable  quantities  are  so  re- 
lated to  each  other  that  the  values  of  one  depend  upon  the  values 
of  the  other. 

Thus,  if  a  train  runs  40  miles  an  hour,  the  distance  that  it  has  traveled 
depends  upon  the  time  elapsed  since  it  started.  As  the  time  increases  the 
distance  increases. 

The  relation  between  two  such  variables  may  be  expressed  by  an 
equation.  \.        « 

Thus,  in  the  above  example,  if  the  distance  in  miles  is  denoted  by  d,  and 
the  time  in  hours  by  t,  then  the  relation  between  them  is  expressed  by 
d  =  iOt. 

One  of  the  simplest  cases  of  related  variables  is  that  in  which 
the  ratio  of  two  variables  is  constant.     If  x  and  y  are  the  variables, 

then  -  =  fc,  or  x  =  Tcy.  where  A;  is  a  constant.     In  this  case  it  is 

y 

said  that  "  x  varies  directly  as  y." 

It  is  evident  that  when  one  quantity  varies  directly  as  another, 
the  ratio  of  any  two  corresponding  values  of  the  variables  equals  the 
ratio  of  any  other  two  con'esponding  values,  since  each  ratio  equals 
the  same  constant. 

Thus,  if  X  varies  directly  as  y,  and  Xi  and  t/i,  and  xg  and  y^  are  any  two 
pairs  of  corresponding  values  of  x  and  y,  then 

X\  _fC2 

Hence,  the  fact  that  x  varies  directly  as  y  is  sometimes  ex- 
pressed by  saying  that  "  a;  is  proportional  to  ?/." 

Example.  — m  varies  directly  as  n,  and  when  m  is  3,  n  is  5.  Write  the 
equation  expressing  the  relation  between  m  and  n.     Find  m  when  n  is  16. 

n      6 
IfnislS,  B=|. 

Solving,  n»  =  9. 


220  ELEMENTARY  ALGEBRA 


EXERCISES 


1.  If  a  train  runs  35  miles  an  hour,  write  the  equation  ex> 
pressing  the  relation  between  the  variable  time  t  and  the  dis- 
tance d. 

2.  If  the  price  of  coffee  is  30  /  a  pound,  the  cost  of  any  quan- 
tity varies  as  the  number  of  pounds  bought.  If  c  =  cost  and 
w  =  weight  in  pounds,  express  the  relation  between  them  by  an 
equation. 

3.  If  the  price  of  sugar  is  5  /  a  pound,  express  by  an  equation 
the  relation  between  the  number  of  pounds  bought  and  the  cost. 

4.  If  the  rate  of  interest  is  6  %,  the  amount  of  interest  on 
any  sum  of  money  varies  as  the  time.  Express  by  an  equation 
the  relation  between  the  interest  and  the  time. 

5.  The  velocity  of  a  body  let  fall  toward  the  ground  varies  as 
the  time  during  which  it  has  fallen  from  rest,  and  the  velocity 
at  the  end  of  3  sec.  is  96  ft.  per  second.  As  the  time  increases, 
the  velocity  increases  at  the  same  rate.  Write  the  equation 
between  £he  velocity  and  time. 

6.  The  force  with  which  a  moving  body  of  any  given  velocity 
strikes  a  stationary  body  varies  as  the  mass  or  weight  of  the 
moving  body.  If  I  strike  a  nail  with  a  force  of  15  lb.  by  using  a 
hammer  weighing  i  lb.,  with  what  force  would  a  2  lb.  hammer 
strike  it  when  swinging  with  the  same  speed  ? 

7.  The  resistance  offered  by  a  wire  of  given  size  to  a  current 
of  electricity  varies  directly  as  the  length  of  the  wire.  If  a  wire 
50  ft.  long  gives  a  resistance  of  6  ohms,  what  will  be  the  resist- 
ance of  75ft.? 

8.  If  R  is  the  resistance  in  Problem  7  and  L  the  length  of 
wire,  write  the  equation  between  them. 

9.  If  X  varies  directly  as  y,  and  x=2  when  y  =  9,  find  x  when 
y  =  36.     Write  the  equation  between  x  and  y. 

10.   If  P  varies  directly  as  V,  and  P=  7  when  V=  5,  write  the 
equation  between  P  and  V.     Find  V  when  P  =  35. 


PROPORTION.     VARIABLES 


221 


122.  Variation  shown  by  Graphs.  —  In  dealing  with  variable 
quantities,  their  variation  often  is  depicted  to  the  eye  by  means 
of  a  chart  or  graph.     See  §  29. 

For  example,  the  public  debt  of  the  United  States,  in  millions  of  dollars, 
from  1820  to  1900  is  given  in  the  following  table  : 


Year 

1820 

1830 

1840 

1850 

I860 

1866 

1870 

1880 

1890 

1900 

Debt 

91 

49 

4 

63 

65 

2773 

2480 

2120 

1652 

2136 

On  squared  paper  draw  a  horizontal  line  OX  and  a  line  OF  at  right  angles 
to  it.     These  are  called  the  axes.    The  point  0  is  called  the  origin. 


2500 
2000 
1500 
1000 
500 


^^ 


1820       1830       1840       1850       I860       1870       1880       1890       1900 


On  the  line  OX  let  a  distance  of  one  space  represent  2  years  of  time. 
Then  the  dates  in  the  above  table  will  be  represented  on  OX  as  in  the  chart. 
On  OF  let  a  distance  of  one  space  represent  100  million  dollars.  On  the 
1820  vertical  line  measure  off  a  distance  representing  91  million  dollars,  and 
mark  a  point.  On  the  1830  vertical  line  measure  off  a  distance  representing 
49  million  dollars,  and  mark  a  point.  Similarly,  locate  a  point  on  each  of 
the  other  date  lines  corresponding  to  the  amount  of  the  debt  of  that  date. 
Draw  a  line  connecting  all  of  the  points  obtained. 

This  curved  line,  called  the  graph  of  the  national  debt,  depicts  clearly  to 


22£ 


ELEMENTARY  ALGEBRA 


the  eye  the  variation  of  the  national  debt  throughout  the  period  of  time 
considered. 

Note.  —  Various  kinds  of  apparatus  are  made  for  constructing  graphs 
automatically.  The  thermograph  or  recording  thermometer  is  a  good  illus- 
tration. A  pen  connected  to  a  thermometer  moves  up  and  down  as  the 
temperature  rises  and  falls.  At  the  same  time  a  clock  mechanism  runs  a 
strip  of  ruled  paper  under  the  pen,  so  that  the  pen  traces  a  continuous  curve 
or  graph  on  the  paper. 

Among  other  instruments  for  making  graphs  is  the  seismograph^  used  for 
recording  earthquakes. 

The  pupil  should  be  provided  with  several  sheets  of  squared 
paper  for  use  in  the  exercises  of  this  and  subsequent  chapters. 

EXERCISES 

1.  The  number  of  teachers  in  the  public  schools  of  Chicago 
since  1850  has  been  as  follows : 


Year 

1850 

1860 

1870 

1880 

1890 

1900 

1909 

Teachers 

21 

123 

557 

898 

2711 

5806 

6296 

Draw  a  graph  showing  the  growth  in  the  size  of  the  teaching 
force  during  that  time. 

Suggestion.  —  On  squared  paper  draw  axes  as  in  the  example  above.  On 
one  represent  the  time,  letting  one  space  represent  2  years.  On  the  other 
represent  the  number  of  teachers,  letting  one  space  represent  200  teachers. 

2.  The  following  average  heights  of  children  at  different  ages 
have  been  determined  from  measurements  of  thousands  of  indi- 
viduals : 


Age 

JYr. 

1  Yb. 

2  Yr. 

3  Ye. 

4  Ye. 

5  Yb. 

6  Ye. 

7  Yr. 

8  Yr. 

Height 

24  in. 

29  in. 

36  m. 

40in. 

42  in. 

44  in. 

46  in. 

48  in. 

50  in. 

Draw  a  graph  showing  the  rate  of  growth  of  the  average  indi- 
vidual. 


PROPORTION.     VARIABLES 


223 


3.  The  population,  to  the  nearest  million,  of  the  United  States 
since  1800  is  given  by  decades  as  follows : 

1800,  5  1840,  17  1880,  50 

1810,  7  1850,  23  1890,  63 

1820,  10  1860,  31  1900,  76 

1830,  13  1870,  39  1910,  93 

Draw  a  graph  showing  the  growth  of  the  population. 

4.  The  population,  to  the  nearest  thousand,  of  the  city  oi 
Chicago  siuce  18.40  is  recorded  as  follows : 

1840,     4  1860,   109  1880,     503  1900,   1699 

1850,   28  1870,   299  1890,   1100  1910,   2185 

Draw  a  graph  showing  the  growth  of  the  city. 

5.  The  following  table  gives,  in  billions  of  dollars,  the  total 
value  of  the  farm  property  and  products  of  the  United  States  by 
decades  from  1850  to  1900: 


Year 

1850 

I860 

1870 

1880 

1890 

1900 

Value 

4 

8 

11 

12  ' 

16 

21 

Show  by  a  graph  the  growth  of  agriculture  in  the  United  States. 

6.  The  amount  of  the  annual  exports  of  the  United  States,  in 
millions  of  dollars,  has  been  as  follows : 

1790,  20  1820,     70  1850,   152  1880,     853 

1800,   71  1830,     74  1860,   400  1890,     910 

1810,   67  1840,   132  1870,   451  1900,   1499 

Show  the  growth  of  our  exports  by  a  graph. 

7.  The  temperature  at  a  place  was  recorded  as  follows :  6  a.m.  . 
8°;  7  A.M.,  8°;  8  a.m.,  10°;  9  a.m.,  11°;  10  a.m.,  14°;  11  a.m.,  18°; 
12  m.,  20°;  1  p.m.,  20°;  2  p.m.,  20°;  3  p.m.,  18°;  4  p.m.,  16°;  5  p.m., 
15° ;  6  p.m.,  13°.  Draw  a  graph  to  show  the  variation  of  tempera^ 
ture  during  the  12  hours. 


224 


ELEMENTARY  ALGEBRA 


123.  Price  Curves,  etc.  —  Graphs  may  be  constructed  and  used 
for  determining  costs  of  different  quantities  of  goods,  interest  on 
money  for  different  periods  of  time,  etc.,  without  computation. 
This  is  shown  by  an  example. 

If  eggs  sell  at  30  ^  a  dozen,  the  relation  between  the  number  of  dozens  and 
the  cost  may  be  expressed  by  the  equation 

c  =  30c?, 

where  d  is  the  number  of  dozens  and  c  is  their  cost.    If  values  are  given  to  d, 
corresponding  values  may  be  found  for  c,  as  given  in  the  table : 


d 

0 

2 

4 

6 

8 

10 

12 

G 

0 

60 

120 

180 

240 

300 

360 

On  squared  paper  draw  two  axes,  OX  and  0  Y,  at  right  angles,  as  in  §  122. 


,-"' 

-  -u      - 

.» ' 

' ' 

*»  ' 

t                         ^  "^  ' 

■   e        ^^^' 

, "" 

^ 

■    I* 55"-    -j^ 

\kd          aid          p<bd     ^ 

X        - 

_      .           -±      -. 

■\  — 

On  OF  let  a  space  represent  1  doz.,  and  on  OX  let  a  space  represent  10 J^. 
Then,  on  the  60  j^  line  mark  a  point  representing  2  doz.  On  the  120  j^  line 
mark  a  point  representing  4  doz.,  etc.  Draw  a  line  through  the  points 
thus  marked.  It  is  seen  that  this  line  or  graph  is  a  straight  line.  It  is  called 
the  "  price  curve."     Most  price  curves  are  straight  lines. 

By  looking  at  this  price  curve  we  can  get  the  cost  of  any  number  of  dozens, 
even  of  a  fractional  number.  For  example,  to  find  the  cost  of  7  doz.,  observe 
the  point  where  the  horizontal  line  7  spaces  up  meets  the  price  curve  ;  observe 
the  point  directly  beneath  this  on  the  axis  OX;  this  is  21  spaces  from  0,  and 
hence  represents  $2.10.    Similarly,  the  cost  of  9^  doz.  is  seen  to  be  $2.85. 


PROPORTION.     VARIABLES  225 

EXERCISES 

1.  Since  the  price  curve  in  the  example  in  §  123  is  a  straight 
line,  how  many  of  the  points  would  have  to  be  located  through 
which  to  draw  it  ?  Should  these  be  taken  close  together  or 
far  apart,  in  order  to  get  the  position  of  the  price  curve  most 
exact  ? 

2.  From  the  price  curve  in  §  123,  give  the  costs  of  the  follow- 
ing :  3  doz. ;  5  doz. ;  10^  doz. ;  6  j  doz. ;  8|^  doz. 

3.  If  eggs  sell  at  25  ^  a  dozen,  draw  the  price  curve. 

4.  On  this  price  curve,  find  the  cost  of  11  doz. ;  of  7^  doz. ;  of 
9f  doz. 

5.  If  coffee  sells  at  28^  a  pound,  draw  the  price  curve. 

6.  On  this  price  curve  find  the  cost  of  8^^  lb. ;  17  lb. ;  24  lb. 

7.  If  sugar  sells  at  5|/  a  pound,  draw  the  price  curve. 

8.  On  this  price  curve  find  the  cost  of  16  lb. ;  of  42  lb. ;  of 
8|-  lb. 

9.  If  money  is  loaned  at  4  %  interest,  draw  a  graph  showing 
the  interest  on  $  1  for  different  periods  of  time. 

Suggestion.  —  Let  distances  on  one  axis  represent  time,  a  space  denoting 
a  month  ;  and  let  distances  on  the  other  axis  represent  interest  on  §  1,  a  space 
denoting  1^. 

10.  On  this  graph  find  the  interest  on  $  1  for  18  mo. ;  for  9  mo. ; 
for  22  mo. ;  for  3  yr.  5  mo. ;.  for  5  yr.  2  mo. 

11.  On  the  graph  in  Problem  9  find  the  interest  on  $450  at 
4  %  for  15  mo. 

Suggestion.  — Find  the  interest  on  $  1,  and  multiply  this  by  450. 

12.  On  the  graph  in  Problem  9  find  the  interest  on  $275  at 
4  %  for  1  yr.  7  mo. ;  on  $  1260  at  4  %  for  3  yr.  10  mo. 

13.  If  money  is  loaned  at  6  %,  draw  a  graph  showing  the  in- 
terest on  $  1  for  different  periods  of  time.  Use  this  to  find  the 
interest  on  different  sums  of  money  at  6  %  for  different  lengths 
of  time. 


226 


ELEMENTARY  ALGEBRA 


14.  The  relation  between  the  circumference  of  a  circle  and  the 
diameter  is  expressed  by  c  =  rrd,  where  c  denotes  the  circumference 
and  d  the  diameter.  Using  tt  =  3^,  draw  the  graph  expressing 
this  relation.  On  this  graph  give  at  sight  the  approximate  values 
of  the  circumferences  of  circles  whose  diameters  are  4  in, ;  5\  in. ; 
8f  in. 

15.  The  area  of  a  circle  is  found  from  A  =  ttE^j  where  A  de- 
notes the  area  and  M  the  radius.  Using  tt  =  3|,  a  few  values  of 
A  for  corresponding  values  of  li  are  given  in  the  table : 


R 

0 

1 

2 

3 

4 

A 

0 

H 

12f 

28f 

50| 

Representing  values  of  the  radius  B  on  the  axis  0  Y,  one  space 
denoting  1  in.,  and  values  of  the  area  A  on  the  axis  OX,  one  space 
denoting  1  sq.  in.,  the  graph  showing  the  relation  between  the 


-^ 

~ 

Y 

«o 

:> 

n 

< 

Cf 

_ 

_^ 

_ 

- 

- 

^ 

- 

' 

r 

- 

_ 

— 

— 

— 

— 

■ 

^ 

»* 

. 

0 

> 

.F 

E 

A 

; 

_ 

_ 

_ 

_ 

L 

U 

L 

radius  and  the  area  is  found  to  be  the  curve  in  the  figure.  The 
approximate  value  of  the  area  of  a  circle  with  any  given  radius 
or  of  the  radius  of  a  circle  with  any  given  area,  may  be  deter- 
mined at  sight  by  this  graph. 

Give  at  sight  the  area  of  a  circle  whose  radius  is  1|  in. ;  21  in. 

Give  at  sight  the  radius  of  a  circle  whose  area  is  10  sq.  in. 

Note.  —  Observe  that  the  graph  in  Problem  16  is  not  a  straight  line.  Many- 
problems  lead  to  graphs  that  are  not  straight  lines.  Some  of  these  will  be 
encountered  in  more  advanced  work. 


PROPORTION.     VARIABLES 


227 


124.  Graphs  of  Linear  Equations.  —  As  shown  in  the  preceding 
sections,  the  relation  between  two  quantities  may  be  expressed  by 
an  equation,  also  the  relation  may  be  shown  by  a  graph.  The 
graph  is  often  spoken  of  as  the  graph  of  the  equation. 

The  two  literal  quantities  in  an  equation  may,  in  some  cases, 
represent  negative  as  well  as  positive  values.  The  process  of 
drawing  the  graph  of  an  equation  between  two  quantities  which 
have  both  positive  and  negative  values  will  be  given  in  this  section. 
The  construction  of  such  a  graph  is  given  below. 


X 

2 

4 

5 

7 

10 

1 

0 

-2 

-6 

-7 

y 

0 

4 

6 

10 

16 

-2 

-4 

-8 

-14 

-18 

Let  the  equation  be  2ic  =  y  +  4.  By  assigning  values  to  y,  compute  the 
corresponding  values  of  x  and  tabulate  the  results  as  shown  in  the  table. 
Thus,  when  y  =  0,  the  equation  becomes 
2  x  =  4,  from  which  a;  =  2.  When  y  =  Qt 
2  a;  =  10,  from  which  x  =  5.  When 
y=—  8,  2aj  =  —  4,  from  which  a;  =  —  2, 
etc. 

Two  axes,  XX^  and  YY',  are  drawn 
at  right  angles  and  meeting  at  0.  Cor- 
responding to  each  set  of  values  of  x  and 
y,  a  point  is  located,  as  in  §  122  and 
§  123,  the  values  of  x  being  measured 
along  or  parallel  to  XX,  and  the  values 
of  y  along  or  parallel  to  YY'.  Positive 
values  of  x  are  measured  to  the  right  of 
YY'  and  negative  values  to  the  left,  as  in 
§  28.  Positive  values  of  y  are  measured 
above  XX  and  negative  values  below 
XX.  For  example,  the  point  A  corre- 
sponding to  X  =  4  and  ?/  =  4  is  obtained 
by  measuring  4  spaces  to  the  right  and  4 
spaces  upward.  The  point  B  correspond- 
ing to  X  =  1  and  y  =  —  2  is  obtained  by 
measuring  1  space  to  the  right  and  2 
spaces  downward.  The  point  C  corre- 
sponding to  X  =  —  2  and  y  =  —  8  is 
obtained  by  measuring  2  spaces  to  the 


:a^ 


J 


?5 


228  ELEMENTARY  ALGEBRA 

left  and  8  spaces  downward.  The  point  D  corresponding  to  x  =  2  and 
y  =  0  is  obtained  by  measuring  2  spaces  to  the  right  on  XX' ;  etc.  By  draw- 
ing a  line  through  all  of  the  points  located  the  graph  of  the  equation  is 
obtained. 

It  is  seen  that  the  graph  of  the  above  equation  is  a  straight 
line.  It  is  shown  in  more  advanced  mathematics,  and  may  be 
assumed  here,  that 

TJie  graph  of  every  linear  equation  containiyig  two  variables  is 
a  straight  line. 

Note.  — It  is  seen  that  in  locating  a  point  corresponding  to  a  set  of  values 
of  the  two  variables  in  an  equation,  the  signs  of  the  numbers  serve  to  tell  the 
directions  of  the  point  from  the  two  axes.  This  is  analogous  to  the  method 
of  locating  a  point  on  the  earth's  surface  by  knowing  its  longitude  and  latitude. 
K  the  latitude  of  a  place  is  minus,  the  place  is  south  of  the  equator,  etc. 


EXERCISES 

Draw  two  axes  and  locate  the  following  points : 

1.  a;  =   4, 2/  =  6.  7.  a?  =  —  6, 2/  =  5. 

2.  a;  =  10,2/  =  7.  8.  a?  =  -  10,  i/ =  12. 

3.  a?  =  15,  2/  =  2.  9.  j»  =  9,  2/  =  —  4. 

4.  a;  =   0, 2/  =  8.  10.  a;  =  15, 2/  =  —  5. 

5.  a;  =  12,  2/=0.  11.  aj  =  -6,  2/  =  -6. 

Q.  x=   0,2/  =  0.  12.  a;  =  -10, 2/  =  -12. 

13.  By  assigning  eight  different  values  to  y  in  the  equation 
a;  —  2  2/  =  7,  including  at  least  two  negative  values,  compute  the 
eight  corresponding  values  of  x.  Draw  axes  and  locate  the  eight 
points  corresponding  to  these  sets  of  values  of  x  and  y.  Are  all 
of  these  points  in  one  straight  line  ?  Draw  a  line  through  all  of 
the  points. 

By  assigning  eight  values  to  y  in  each  of  the  following  equa- 
tions, including  at  least  three  negative  values,  and  computing  the 
corresponding  values  of  x,  locate  eight  points  of  the  graph  of  the 
equation,  and  draw  the  graph. 


PROPORTION,     VARIABLES  229 

14.  x  =  y.  16.   a; +  4  2/ =  10.  18.   5  a;  — 4?/ =  20. 

15.  a;  +  2^  =  0.  17.   ^x-y  =  12.  19.   2x  +  Zy  =  12, 

20.  Since  the  graph  of  a  linear  equation  in  two  variables  is  a 
straight  line,  how  many  points  of  it  must  be  located  in  order  to 
draw  it  accurately  ? 

By  locating  only  two  points  of  each,  draw  the  graphs  of  all  of 
the  following  equations  on  one  pair  of  axes : 

21.  Sx  —  y  =  12.  23.    2  a;  — 4  2/ =  5.  25.  5  a;  — y  =  4. 

22.  y=2x.  24.    2x-^^y  =  l,  26.  ^x  =  ^y. 

27.  If  the  cost  of  setting  the  type  for  printing  a  circular  is 
75^,  and  the  cost  of  paper  and  press  work  in  printing  it  is  ^^  a 
copy,  then 

c  =  |w  +  75, 

where  c  denotes  the  cost  in  cents  of  printing  any  number  of 
copies  and  n  denotes  the  number  of  copies  printed. 

Draw  the  graph  of  this  equation. 

Give  at  sight  from  the  graph  the  cost  of  12  copies ;  20  copies ; 
100  copies. 

SUPPLEMENTARY  EXERCISES 

1.  Express  in  simplest  form  the  ratio  ofm tol • 

2.  Separate  72  into  three  parts  which  are  in  the  ratio  of 
2:3:4. 

Q     I     Q 

3.  If  n  is  a  positive  number,  which  is  the  greater  ratio,  - — - — 

o+4n 

or|±ii;? 

3  +  5w 

Suggestion.  —  Reduce  to  a  common  denominator,  then  compare  the  nu- 
merators. 

4.  If  X  and  y  are  positive  numbers,  which  is  the  greater  ratio, 
a?  +  5.v  ^j.  x±7_y^ 

x  +  6y       x-{-Sy' 


230  ELEMENTARY  ALGEBRA 

5.  Find  the  mean  proportional  between  4  a^  and  y  a{a  ~  by. 

6.  Find  the  fourth  proportional  to  v^  —  w'\  ,  and  v  —  w. 

7.  Write  in  three  ways  the  proportion  between  3,  4,  20,  and  15. 

8.  Write  as  a  proportion  oc^  —  y^  =  5xQ. 

9.  If  £  =  !-=«,  show  that  4±|5^±|^  =  £. 

a'     b'     c'  a'+3b'  +  5c'     a' 

10.  If  2  =  £,  show  that  ^jz^^e^jz^. 

11.  If  2  =  ^,  show  that -5^= i±^ 

h     d  a-{-c     a -\- b -\- c -\- d 

12.  In  the  similar  polygons  ABODE  and  A'B'C'D*E'  it  is 

known  that 

AB       BC       CD       DE       EA 


A'B'     B'C     CD'     D'E'     E'A' 


Show  that  the  ratio  of  the  perimeters  of  the  polygons  equals  the 
ratio  of  any  two  corresponding  sides. 

13.  X  varies  as  y,  and  when  x  =  2,  y  =  5.     Find  y  when  x  —  15. 

14.  The  distance  that  a  body  falls  from  rest  varies  as  the  square 
of  the  time.  In  2  seconds  it  falls  64  ft.  How  far  will  it  fall  in 
3  sec.  ?     In  4  sec.  ?     In  10  sec.  ? 

15.  One  quantity  is  said  to  vary  inversely  as  another  when, 
during  all  of  their  changes,  their  product  remains  constant.  As 
one  increases,  the  other  decreases.  If  x  varies  inversely  as  y, 
then  xy  =  kj  where  fc  is  a  constant. 


PROPORTION.     VARIABLES  231 

If  X  varies  inversely  as  ?/,  and  y=^2  when  a;  =  4,  find  y  when 
a;  =  32. 

16.  The  volume  of  any  gas  varies  inversely  as  the  pressure 
upon  it.  When  the  pressure  is  8  lb.  the  volume  is  8  cu.  in.  What 
is  the  volume  when  the  pressure  is  4  lb.? 

17.  The  number  of  vibrations  made  by  the  pendulum  of  a 
clock  in  a  given  time  varies  inversely  as  the  square  root  of  its 
length.  A  pendulum  39.1  inches  long  makes  one  vibration  in  a 
second.  How  ^ong  must  a  pendulum  be  to  make  4  vibrations  in 
a  second^ 


CHAPTER  XII 
SYSTEMS   OF  LINEAR  EQUATIONS 

125.  Systems  of  Equations. — In  Chapter  VI  it  was  shown  that 
some  problems  may  be  expressed  by  means  of  two  linear  equa- 
tions containing  two  unknown  numbers.  These  equations  were 
called  simultaneous,  and  were  said  to  form  a  system.  It  was  found 
that  the  equations  in  each  system  in  that  chapter  had  one  set  of 
values  of  the  unknown  numbers  that  satisfied  both  equations, 
called  a  solution  of  the  system.  This  solution  was  discovered 
through  a  process  called  elimination. 

There  are  three  principal  methods  of  elimination  in  common 
use  that  will  now  be  discussed,  and  applied  in  the  solution  of 
problems. 

126.  Elimination  by  Addition  or  Subtraction.  —  The  method  of 
elimination  shown  in  §  68  is  known  as  the  method  of  elimination  by 
addition  or  subtraction.  The  student  should  now  study  again  the 
rule  on  page  110.  The  following  exercises  are  given  for  review 
of  that  process. 

TJie  student  should  check  every  answer  by  seeing  if  the  values  of 
the  unknown  numbers  found  satisfy  both  of  the  given  equations. 

EXERCISES 

Eliminate  by  addition  or  subtraction,  solve,  and  check : 

7  s  +  ^  =  42, 


r3a-6  =  21,  {ls  + 

'    l2a  +  &  =  4.  ,            •    \Ss- 

2     rm  +  4?i=4,  5     H 

\m-2n  =  16.  '    [o 

\3A- 


t  =  S. 
w  -{-v  =  25, 
w-2v  =  S5. 
2x-y  =  5,  f4^-3J5  =  l, 

[5x-2y=U.  '    [3A-4.B  =  6. 

232 


'SYSTEMS   OF  LINEAR   EQUATIONS 


233 


7. 


9. 


10. 


11. 


12. 


13. 


14, 


[2x  =  y-6, 
(Sp  +  2q=12, 
[4.p  =  3q-l, 
5R-2r  =  ly 
SE  =  5r-ll, 
r3Jf-2^44  =  0,- 
[2M-N-{-l  =  0. 
(2h-}-k  =  35y 
[5h-3k=2r. 
r  5  2/  —  5  a;  =  15, 
[Sx  +  5y  =  71, 
J3ri  +  2r2  =  4, 
l4r2-3ri  +  l  =  0. 
-4jB  +  3(7  =  46, 
2(7  +  6jB  =  4. 


15, 


16, 


17. 


18. 


19. 


20 


f  m  =  4  —  2  w, 

[2n-\-12  =  m. 


8  W-h  w  =  7, 
W+2w  =  28. 
fl0/4-4«=22, 
Uh-5/=11. 


r  2  a^  —  ajg  =  9, 
1  5  a^i  —  3  ajg  = 


14. 


"-4-^-9 

7      2^ 
6      4       ' 
.4^6       ^ 


Example.  — Solve 
Eliminate  x. 
Solving  (1)  forx, 


j4x  +  y  =  34, 
\  4  J/  +  a;  =  16. 


127.   Elimination  by  Comparison.  —  The  method  of  elimination  by 
comparison  is  illustrated  in  the  following  example. 

(1) 
(2) 

(8) 

(4) 

(6) 


4 

Solving  (2)  for  x,  a;  =  16  -  4  2^. 

Comparing  the  two  values  of  x  given  in  (3)  and  (4), 

4 

Solving  (6)  for  y,  1/  =  2. 

Replacing  y  in  (2)  by  its  value  2, 

8  +  X  =  16. 
Solving,  X  =  8. 

Hence  the  solution  is  x  =  S,  y  =  2. 

This  example  illustrates  the  rule : 

Solve  each  of  the  eqtiations  for  the  value  of  either  one  of  the  un- 
knoim  numhersj  expressed  in  terms  of  the  other ^  and  write  one  of  tlie 


234 


ELEMENTARY  ALGEBRA 


values  equal  to  the  other.  Solve  the  resulting  equation.  Substitute 
the  value  found  for  the  one  unknown  number  in  either  of  the  origiual 
equations,  and  solve  for  the  other  unknown  number. 


EXERCISES 


Eliminate  by  comparison,  solve,  and  check : 
\m--7i=  —  1. 


2     ra;-5a  =  7. 


[2  a; -15  a  =9. 
r  +  2  s  =  4, 
3  r  -  s  =  5. 


-I 

l5ri-3r2  =  : 


14. 


3^  +  25  =  2, 


I 

f3i>- 

(v  +  2t 
\2t~v 


^-^=9. 
g  =  l, 
5g  =  41. 

=  4, 

+  12  =  0. 

9  Jf  -  5  N=  13, 
5Jf+J\^=ll. 
3e-5/=5, 
7e+/=265. 


10    pi^-^=5> 

17?  +  2T=25. 

riO  F+3n=:174, 
^^-    1 3  F+10n  =  125. 
r5a;  +  2aj'  =  l, 
|l3a;  +  8a;'  =  ll. 


13. 


2h-{-k  =  9, 
5h  +  Sk  =  25. 


14.     ^ 


rjB  +  2Q  +  2=0, 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


22. 


8S. 


[4  5-7  Q=  37. 
r  6  2/  -  5  2;  =  1, 
l92/  +  10;2  =  12. 
'7i:  +  3Q  +  9  =  0, 
.  6  Q  -  9  i  =  28. 
3W-{-2D  =  17, 

4  Tr+Z)  =  16. 
8  a  +  5  ^  =  5, 

3  a  -  2  «=  29. 

5  a;  -f  9  2/  =  8, 
6?/-9a;  +  7  =0. 
5F+4P=22, 

3  F+P=9. 
T     5  » 

6  +  3  =  ^' 

2r^o    7^ 


3 

7_E 
4 


19 
15' 


2 


TF- 

7  TF 


3^  3 


SYSTEMS   OF  LINEAR   EQUATIONS       '         235 

128.  Elimination  by  Substitution.  —  The  method  of  elimination 
by  substitution  is  illustrated  in  the  following  example. 

^  e  1       ( 4  to  -  5  w  =  26,  (1) 

Example.  —  Solve  J  ^  '  ^^( 

[3M?-6t;=15.  (2) 

Eliminate  w 

Solving  (1)  for  w,  w=  ?^-±A?.  (3) 

Substituting  — — — -  in  place  of  w  in  (2), 
4 

sn^y\-ev^i6.  (4) 

Solving  (4),  v  =  2. 

Replacing  v  by  its  value  2  in  (1), 

4  w?  -  10  =  26. 
Solving,  w  =  9. 

Hence,  the  solution  is  w  =  9,  ©  =  2. 

This  example  illustrates  the  rule: 

Solve  one  of  the  equations  for  the  value  of  either  of  the  unknown 
numbers,  expressed  in  terms  of  the  otherf  and  substitute  this  value  in 
place  of  that  number  in  the  other  equation.  Solve  the  resulting 
equation.  Substitute  the  value  found  for  the  one  unknown  number 
in  either  of  the  original  equations,  and  solve  for  the  other  un- 
known  number, 

EXERCISES 

Eliminate  by  substitution,  solve,  and  check: 

( 


2. 


[a  +  36  =  9.  1F+3«=16. 

x-y  =  ^,  r3a-7»  =  40, 

a;H-22/  =  16.  \4a-3;2  =  9. 

rm  +  4n=7,  f4p-5g  =  26, 

•    lm  +  6n  =  9.  |3p-6g  =  15. 

ra;i  +  ^2  =  30,  \^D 

[3a^-2a;2  =  25.  ^'     {2  D 

i-2k  = 

33. 


+  7Q  =  16, 
+  5Q  =  13. 


(R-S==^,  f3^  +  2A:  =  26, 

^     [2R  +  S=-14.  *"•    \5g-2k=^ 


236 


ELEMENTARY  ALGEBRA 


11. 


12. 


13. 


14. 


15. 


16. 


17. 


7^-95=13, 
5  ^  +  2  J5  =  10. 

7ri4-4r2  =  l, 
9  ri  +  4  ^2  =  3. 


\i 


T+t  =  10, 


r  7  'u  +  s  =  42, 

2  M-  6  =  5, 
5  Jf- 2  6  =  14. 

r^-8j9  =  45, 
13^-2)  =  20. 

r3a;4-22/  =  26, 
[5a;  =  38  +  2y. 


18. 


19. 


21. 


22. 


r7m  =  26  +  3a, 
I  6  a  =  6  m  -  20. 

r8i/  +  rf 
l72/  =  9 


60, 
20(?. 
ri6-7^=3r, 
|2r-13  +  5^  =  0. 


a+  A;  =30, 

2      3        3' 


-  + 


a— o      6—4 

3  a  =  2  6  +  7. 

r2w;_ 

3 

4:W  -\-v  _2 
6w  -i-v     5 


=  0, 


5, 


MISCELLANEOUS  EXERCISES 

Eliminate  by  the  method  that  seems  best  suited,  and  solve 


3. 


[x-y  =  4:. 


7v  —  10w=zd. 

(21Ii  +  8t-{-66  =  0, 
1 49  jR  +  53  =  15  ^ 

r5Jf+63  =  26, 
l2Jf+6  =  18. 
r2a-26  =  6|, 
l7a  =  264-36. 

r3n  +  152;  =  7, 
^-    1 12 -n +  5  2  =  0. 


10. 


11. 


^-7  +  j-O, 
|  =  9m-5t, 

1+5  =  1 

X     y     xy* 
2x-3y  =  l. 

r2-5P=4  F, 

3^4 
[p-1     v+2' 


SYSTEMS   OF  LINEAR   EQUATIONS 


237 


129.  Problems  solved  by  Systems  of  Equations.  —  For  the  steps 
in  the  process  of  expressing  a  problem  by  means  of  a  system  of 
equations  with  two  unknown  numbers  see  §  69.  In  the  follow- 
ing problems  use  that  method  of  elimination  which  seems  best 
suited  to  each  individual  problem. 

EXERCISES 

1.  Find  two  numbers  whose  sum  is  29,  and  difference  13. 

2.  Find  two  numbers  whose  sum  is  9,  and  such  that  one  of 
them  exceeds  twice  the  other  by  27. 

3.  The  difference  between  two  numbers  is  72,  and  one  of  them 
is  4  times  the  other.     What  are  the  numbers  ? 

4.  The  sum  of  two  numbers  is  64,  and  their  quotient  7.  What 
are  the  numbers? 

5.  The  difference 'between  two  numbers  is  21,  and  if  the  larger 
is  divided  by  the  smaller,  the  quotient  is  4  and  the  remainder  3. 
Find  the  numbers. 

6.  The  value  of  a  fraction  is  J,  and  if  7  be  added  to  each  term, 
the  value  of  the  resulting  fraction  is  J.     Find  the  fraction. 

7.  A  board  10  ft.  long  is  to  be  cut  into  two  pieces  whose 
lengths  are  as  3  and  4.     How  long  are  the  pieces  ? 

8.  A  surveyor  wishes  to  set  a  stake  in  a  line  320  ft.  long,  so  that 
its  distances  from  the  ends  of  the  line  are  as  7  is  to  19.  Where 
must  he  place  it? 

9.  In  building  a  bridge  a  steel  bar  12  ft.  long  is  to  be  bolted  to 
another  bar  at  a  point  which  divides  the  12  ft. 

bar  into  parts  whose  lengths  are  as  7  and  17. 
Where  must  the  hole  be  bored  for  the  bolt  ? 

10.  In  the  steel  frame  in  the  figure  the 
brace  ^(7  is  15  ft.,  AD  is  7  ft.,  and  DB  is  3  ft. 
It  is  desired  to  find  where  to  bore  the  hole  for 

the  bolt  E  in  AC. 


B,propoHion,||  =  ^. 


Find  where  to  locate  E. 


238  ELEMENTARY  ALGEBRA 

y(.  11.  Two  boys  desired  to  weigh  themselves.  They  had  no 
scales,  but  had  a  10  lb.  weight.  They  placed  a  board  across  a  sup- 
port, as  in  playing  teeter.  When  they  sat,  one  5  ft.  from  the  ful- 
crum and  the  other  4  ft.  from  it,  they  balanced.  When  the  smaller 
boy  took  the  10  lb.  weight  in  his  hands,  he  had  to  sit  only  4  ft. 
4  in.  from  the  fulcrum  to  balance.     What  were  their  weights  ? 

12.  Two  weights  balance  when  the  larger  is  3  ft.  and  the 
smaller  4  ft.  from  the  fulcrum.  If  the  smaller  weight,  with  3  IJ). 
added  to  it,  is  moved  to  a  position  3  ft.  5^  in.  from  the  fulcrum, 
they  still  balance.     Find  the  weights. 

13.  Two  weights  balance  when  one  is  4  in.  and  the  other  6  in. 
from  the  fulcrum.  If  the  first  weight  is  decreased  12  lb.,  the 
other  must  be  moved  1  in.  nearer  the  fulcrum  to  balance.  Find 
the  weights. 

14.  Two  weights  balance  when  one  is  3  ft.  and  the  other  2  ft. 
from  the  fulcrum.  The  larger  weight  is  decreased  12  oz.,  and  the 
smaller  weight  is  decreased  6  oz.  and  moved  2|  in.  nearer  the  ful- 
crum.    Again  they  balance.     Find  the  weights. 

15.  Two  partners  in  business,  A  and  B,  are  to  divide  profits  in 
the  ratio  of  2  to  5.  In  dividing  a  profit  of  $  6300,  how  much 
should  each  receive? 

16.  A  man  invests  part  of  $  12,000  at  5  %  and  the  balance  at 
6%.  His  annual  income  from  both  investments  is  $645.  Find 
the  number  of  dollars  in  each  investment. 

17.  A  man  invests  $2000,  part  at  5%  .and  the  rest  at  4%. 
The  annual  income  from  the  5  %  investment  exceeds  that  of  the 
4%  investment  by  $2.80.     Find  the  amount  of  each  investment. 

18.  An  investor  purchased  two  kinds  of  bonds.  One  kind 
yielded  3%  and  the  other  4%.  His  yearly  income  from  both 
was  $  558.  Had  he  invested  as  much  in  4  %  bonds  as  in  3  % 
bonds,  and  vice  versa,  his  yearly  income  would  have  been  $576. 
How  much  did  he  invest  in  each  kind  of  bonds  ? 

19.  A  man  invested  in  $22,500  worth  of  bonds,  part  of  them 
New  York  City  4s  and  part  U.  S.  Steel  5s.     His  annual  income 


SYSTEMS   OF  LINEAR   EQUATIONS  239 

was  $  1045.  Had  he  purchased  as  many  New  York  City  4s  as 
he  did  U.  S.  Steel  5s,  and  vice  versa,  his  annual  income  would 
have  been  only  $980.  What  was  the  value  of  each  kind  of  bonds 
bought? 

20.  I  change  $3  into  dimes  and  nickels.  There  are  50  coins 
in  all.     How  many  dimes  and  how  many  nickels  are  there  ? 

21.  I  got  $100  changed  into  $5  bills  and  $10  bills.  I  re- 
ceived 16  bills  in  all.     How  many  bills  of  each  kind  did  I  get? 

22.  If  a  merchant  blends  25-cent  coffee  and  32-cent  coffee  to 
sell  at  30  cents  a  pound,  what  quantities  of  each  grade  of  coffee 
must  he  take  to  make  50  lb.  of  the  blend  ? 

^X  23.  How  much  each  of  copper,  specific  gravity  8.9,  and  zinc, 
specific  gravity  6.9,  must  be  combined  to  produce  1000  cu.  cm.  of 
brass,  specific  gravity  8.4  ?     (See  Problem  7,  page  112.) 

24.  In  a  number  of  two  digits  the  sum  of  the  digits  is  11,  and 
when  the  digits  are  interchanged,  the  number  is  diminished  by 
45.     Find  the  number. 

25.  Two  angles  are  called  supplementary  when  their  sum  is 
180°.  If  one  of  two  supplementary  angles  is  20°  more  than  twice 
the  other,  how  many  degrees  in  each  ? 

26.  The  sum  of  the  angles  of  a  triangle  is  always  180°.     If  each 
of  the  angles  at  the  base  of  the  triangle  is  twice  the 
angle  at  the  vertex,  how  many  degrees  in  each  angle 
of  the  triangle  ? 

27.  The  altitude  of  a  trapezoid  is  6  ft.  and  its  area 
84  sq.  ft.  One  base  is  8  ft.  longer  than  the  other. 
Find  the  lengths  of  the  bases. 

28.  If  the  length  of  a  rectangle  is  diminished  3  in.  and  its 
width  increased  2  in.,  its  area  is  unchanged;  and  if  its  length  is 
increased  5  in.  and  its  width  diminished  2  in.,  its  area  is  un- 
changed.    Find  the  length  and  width. 

29.  Two  groups  of  students  of  surveying  were  sent  to  find  the 
area  of  a  rectangular  field.  One  group  got  the  length  4  ft.  too 
small  and  the  width  6  ft.  too  large,  which  gave  the  area  2960  sq. 


240  ELEMENTARY  ALGEBRA 

ft.  too  large.  The  other  group  got  the  length  3  ft.  too  large  and 
the  width  3  ft.  too  small,  which  gave  the  area  741  sq.  ft.  too 
small.     Find  the  dimensions  of  the  field. 

30.  The  circumference  of  the  fore  wheel  of  a  carriage  is  3  ft. 
less  than  that  of  the  rear  wheel.  One  goes  the  same  distance  in 
40  revolutions  that  the  other  does  in  30.  Find  the  circumference 
of  each. 

31.  A  belt  runs  over  two  pulleys.  One  makes  10  revolutions 
while  the  other  makes  3.  If  the  larger  pulley  were  replaced  by 
one  whose  circumference  was  20  in.  less,  it  would  only  make  10 
revolutions  while  the  other  made  4.  Find  the  circumference  of 
each  pulley.  • 

32.  A  man  who  can  row  6  mi.  an  hour  downstream  can  row 
2  mi.  an  hour  upstream.     What  is  the  speed  of  the  current? 

33.  A  crew  rows  downstream  6|  mi.  in  an  hour,  and  returns  in 
4  hr.  20  min.  What  is  the  speed  of  the  current,  and  at  what  rate 
could  the  crew  row  in  still  water  ? 

34..  An  aviator  flying  against  the  wind  makes  20  mi.  an  hour. 
On  returning  with  the  wind  he  makes  50  mi.  an  hour.  What  is 
the  speed  of  the  wind,  and  what  would  be  his  speed  if  there  were 
no  wind  ? 

35.  If  an  aviator  flies  to  a  point  24  mi.  distant,  against  the 
wind,  in  40  min.,  and  returns  in  30  min.,  what  is  the  speed  of  the 
wind,  and  what  would  be  his  speed  if  there  were  no  wind  ? 

36.  Two  trains,  each  300  ft.  long,  run  on  parallel  tracks.  If 
running  in  the  same  direction,  it  requires  20  sec.  for  one  to  pass 
the  other.  If  running  in  opposite  directions,  it  requires  only  4  sec. 
for  them  to  pass.     What  are  the  speeds  of  the  trains? 

37.  A  passenger  train  and  a  freight  train  run  in  opposite  direc- 
tions on  parallel  tracks.  The  speed  of  the  passenger  train  is  60 
ft.  per  second,  and  the  speed  of  the  freight  train  is  40  ft.  per  sec- 
ond. The  freight  train  is  180  ft.  longer  than  the  passenger  train. 
It  takes  the  trains  12.6  sec.  to  pass.     Find  the  length  of  each  train. 


SYSTEMS   OF  LINEAR   EQUATIONS  241 

38.  A  passenger  and  a  freight  train  run  on  parallel  tracks  and 
in  the  same  direction.  The  speed  of  the  passenger  train  is  45 
mi.  an  hour,  and  the  speed  of  the  freight  train  is  25  mi.  an  hour. 
The  freight  train  is  96  ft.  longer  than  the  passenger  train.  It 
takes  the  passenger  train  36  sec.  to  pass  the  freight.  Find  the 
length  of  each  train. 

39.  How  much  milk  testing  4  %  butter  fat  and  cream  testing 
24%  butter  fat  must  be  mixed  to  make  20  gal.  that  test  20% 
butter  fat  ? 

40.  How  much  milk  testing  3.8  %  butter  fat  and  cream  testing 
26.6  %  butter  fat  must  be  mixed  to  make  16  gal.  that  test  18  % 
butter  fat? 

41.  Fifteen  pounds  of  tin  weigh  13  lb.  in  water  and  15  lb.  of 
zinc  weigh  13.5  lb.  in  water.  How  much  tin  and  how  much  zinc 
in  an  alloy  which  weighs  66  lb.  in  air  and  49  lb.  in  water  ? 

42.  Some  authorities  claim  that  the  daily  ration  for  the  aver- 
age adult  workingman  should  contain  4  oz.  of  protein  and  an 
equal  amount  of  fat.  White  bread  contains  9  %  protein  and  1  % 
fat.  Mutton  contains  14%  protein  and  37%  fat.  Find  how 
many  ounces  each. of  bread  and  mutton  would  be  required  to  make 
a  daily  ration. 

43.  Eggs  contain  13%  protein  and  9%  fat.  From  the  facts 
given  in  Problem  42,  can  bread  and  eggs  be  used  to  make  a  stan- 
dard ration  ? 

44.  A  society  that  wished  to  raise  $  100  gave  an  entertainment. 
They  estimated  that  enough  children  and  adults  would  attend 
that  they  could  raise  this  amount  by  charging  children  10  cents 
and  adults  25  cents  admission,  and  that  if  they  charged  children 
15  cents  and  adults  30  cents  admission,  they  would  raise  $26 
more   than  the   amount   desired.     How   many 

children  and  how  many  adults  did  they  expect  C 

would  attend  ? 

45.  In  the  triangle  ABC,  the  line  CD  bisects 
(divides  into  two  equal  parts)  the  angle  at  C.     ^rn      I  n 
It  is  known  from  geometry  that  VD  divides  AB   a 


242 


ELEMENTARY  ALGEBRA 


into  two  parts,  m  and  n,  which  are  proportional  to  the  adjacent 
sides.  If  AC  =  15  in.,  BC  =  12  in.,  and  AB  =  10  in.,  find  the 
lengths  of  m  and  n. 


Example.  —  Solve  for  x  and  y 

Multiplying  (1)  by  6, 
Multiplying  (2)  by  a, 
Subtracting  (4)  from  (3), 
Factoring  (combining  terms), 
Dividing  by  a^  _  52^ 
Substituting  ah  for  y  m  (1), 


130.  Systems  of  Literal  Equations.  —  Systems  of  linear  equations 
in  which  one  or  more  of  the  known  numbers  are  literal  are  solved 
by  the  methods  of  §§  126,  127,  128,  the  solutions  being  expressed 
in  terms  of  these  literal  quantities. 

ax  —  by  =  a\  (1) 

bx  —  ay  =  63.  (2) 

abx  -  h^y  =  a^h.  (3) 

ahx  —  a^y  =  db^.  (4) 

a'^y  -  62y  =  a%  -  ab^. 
ia^-b^)y  =  abia^-b^), 
y  =  ab. 
ax  —  ab^  =  a^ 

ax  =  a^  +  ab\ 
x  =  a^+b^, 
:a^  +  b^,y  =  ab. 


Hence,  the  solution  is 


Solve  for  x  and  y  : 

hx  -\-  ay=2  ah. 
ax  -\-  hy  =  a^  —  p. 


2. 


3. 


4. 


I 

\ay  -\-hx  =  a^  —  h\ 
(  my  -{-  x  =  m  +  rif 
\  mx  +  my  =  m^  -\- n, 
(x-\-Ey  =  S, 
\2x-Sy  =  M. 

Ax  +  y  =  B, 

Bx  +  y  =  A. 
\x-p^y  =  0, 
[X-\-q'y  =  l, 


EXERCISES 

7. 

8.  . 

9. 
10. 
11. 


12. 


rix-i-r2y  =  l, 

r^  +  ^i3^  =  1. 

ax  —  b^y  =  a^  —  h\ 

Q?y  —  hx  =  0. 

2pa;4-2g2/  =  4y  +  g«, 

x  —  2y  =  2p  —  q. 

12x-lly  =  M^  +  12R2 

x-hy  =  2R^  +  B,. 

(m  +  n)x  =  l  —py, 
(m-\-n)y+px  =  l, 
vx  —  ty  =  0, 
x-\-y  =  w. 


SYSTEMS   OF  LINEAR   EQUATIONS 


243 


13. 


14. 


15. 


Px- 

Qy  =  B, 

x^y 

=  Q- 

2^3 

=  a, 

3^4 

=  6. 

X 

4-     ^ 

a-b 

a  +  6 

x-\-y 

=  2a. 

16. 


17. 


X     o        m  —  12n 
Sx  +  9y  =  4:m-\-9n. 


X 
V 

V 


+  5  =  21, 

V 

7y^3^ 
4i;     4* 


18.  If  m  be  added  to  the  numerator  of  a  certain  fraction,  the 
value  of  the  resulting  fraction  is  4.  But  if  n  be  added  to  the 
denominator,  the  value  of  the  resulting  fraction  is  3.  What  is 
the  fraction  ? 

19.  The  sum  of  two  numbers  is  s,  and  the  quotient  of  the  first 
divided  by  the  second  is  q.     Find  the  numbers. 

20.  If  a  boy  who  weighs  IF  pounds  and  one  who  weighs  P  pounds 
balance  at  the  ends  of  a  teeter  board  I  feet  long,  find  the  lengths 
of  the  two  parts  into  which  the  board  is  divided  at  the  fulcrum. 

21.  Part  of  $12,000  is  invested  at  «%  and  the  rest  at  y%. 
The  annual  income  from  both  investments  is  d  dollars.  Find  the 
number  of  dollars  in  each  investment. 

22.  If  milk  tests  m%  butter  fat  and  cream  n<fo  butter  fat, 
how  much  of  each  must  be  taken  to  make  w  pounds  of  cream 
testing  a%  butter  fat? 

131.  Number  of  Solutio»s  of  a  System. — It  has  been  seen  that 
each  of  the  problems  in  the  preceding  sections  has  one  and  only 
one  solution.  That  this  is  true  in  general  of  a  system  of  two 
linear  equations  with  two  unknown  numbers  may  be  shown  as 
follows : 

Any  two  linear  equations  whose  unknown  numbers  are  x  and  y 
may  be  written  in  the  forms 

ax-{-by  =  Cy 
dx  +  ey  =  fy 


244  ELEMENTARY  ALGEBRA 

where  a,  h,  c,  d,  e,  and  /  represent  any  numbers  whatever.     If 
this  general  system  be  solved, 

ce  —  bf  af—  cd 

x  = '—,  y=— 

ae  —  bd  ae  —  bd 

Hence,  in  general,  the  system  has  one  and  only  one  solution, 
depending  upon  the  values  of  a,  b,  c,  d,  e,  and  /.  But  these  num- 
bers may  be  so  related  that  there  is  no  solution,  or  that  there  is 
an  infinitely  great  number  of  solutions. 

Example  1.  -  Solve    f  2  ^  -     2/  =  12,  (1) 

\  6  jc  -  3  y  =   4.  (2) 

Multiplying  (1)  by  3,     6  x  -  3  ?/  =  36.  (3) 

Subtracting  (2)  from  (3),  0  =  32. 

This  is  absurd.  It  is  impossible  to  eliminate  one  of  the  unknown  numbers 
without  at  the  same  time  eliminating  the  other,  and  hence  no  solution  can 
be  found.  Or,  looking  at  the  matter  in  a  different  way,  the  supposition  that 
both  equations  can  be  true  simultaneously,  that  is,  for  the  same  values  of  x 
and  y,  leads  to  the  conclusion  that  0  —  32,  which  is  false.  Hence  the  sup- 
position is  false,  and  the  equations  have  no  common  solution. 

Two  linear  equations  having  no  common  solution  are  called 
inconsistent  equations. 

Example  2.  —  Solve   f  2  x  -  y  =  3,  (1) 

\4.x  =  Q  +  2y.  (2) 

Transposing  (2)  gives             4  a;  —  2  y  =  6.  (3) 

Dividing  (3)  by  2,                      2  x  -  2/  =  3.  (4) 

Equation  (1)  is  the  same  as  equation  (4).  Hence  all  of  their  solutions 
are  common. 

Two  equations,  such  as  (1)  and  (2)  above,  that  have  all  of  their 
solutions  common  are  called  equivalent •  equations.  It  is  evident 
that  one  of  the  equations  may  always  be  derived  from  the  other 
by  transposition,  etc. 

132.  Systems  solved  Graphically.  —  The  solution  of  a  system  of 
two  equations  in  two  unknown  numbers  may  be  found  graphically, 
without  performing  the  process  of  elimination. 

Example.  —  Solve  graphically    l^+    ^^    ' 


SYSTEMS   OF  LINEAR   EQUATIONS 


245 


.    Upon  the  same  pair  of  axes  draw  the  graphs  of  both  equations,  as  in  §  124. 

Only  two  points  of  each  graph  need 

be  located  in  order  to  draw  it.     The 

graph  of  X  +  2  2/  =  4  is  AB,  and  the 

graph  of  2  X  -  y  ==  28  is  CD.     The 

graphs  AB  and  CD  meet  at  a  point 

P.     Since  P  is  on  AB^  the  values 

of  X  and  y   corresponding  to    the 

point    P  must   satisfy   x  +  2  y  =  4. 

And  since   P  is  also   on    CD,  the 

values  of  x  and  y  corresponding  to 

the     point    P    must     also    satisfy 

2  X  —  y  =  28.     Hence  the  values  of 

X  and  y  corresponding  to  the  point 

P,  which  are  seen  to  be  x  =  12,  y  =  —  4,  must  be  the  solution  of  the  system. 

See  if  X  =  12,  y  =  —  4  satisfies  both  equations. 

To  solve  a  system  graphically,  draw  the  graphs  of  both  equations 
upon  the  same  axes.  The  set  of  values  of  the  unknown  numbers 
corresponding  to  the  point  where  the  graphs  meet  is  the  solution  of 
the  system. 

EXERCISES 


:^ :;:;:;:;  i^rpi:  yn:  :p: 


Solve  graphically: 
1. 


2. 


3. 


4. 


5. 


6. 


\x—y=L 

\7x-2y  =  S, 

|6a;- ?/  =  4. 
'»  +  42/  =  10, 
Sx-y  =  12. 

r4a;  +  2^  =  3, 
l2a;  +  32/  =  4. 
r  a;  +  2/  =  5, 
\2x-\-3y  =  12. 


7. 


8. 


10. 


11, 


12. 


r3a;  +  2y  =  12, 
|4a;-32/+l  =  0. 
|7a;-22/  +  15  =  0, 
\y-2x  =  Q. 
r2x  +  52/  =  15, 
|3a;-42/  =  ll. 
r4a;-3y  =  l, 
[3a;-42/  =  6. 
{ 2x  —  y  =  5y 
|a;H-2y  =  25. 
I2x-y  =  b, 
|5a;-2y  =  14. 


13.    Show  by  graphs  that  in  general  a  system  of  two  equations 
in  two  unknown  numbers  has  one  and  only  one  solution. 


246  ELEMENTARY  ALGEBRA 

14.  Draw  the  graphs  of  the  inconsistent  equations  of  the  system 

What  kind  of  lines  are  they?  Would  you  naturally  expect 
this  ? 

15.  Show  graphically  that  the  following  equations  are  in- 
consistent : 

ra:-32/=6, 
l3a;  =  4+9y. 

16.  Determine  graphically  whether  or  not  the  following  equa- 
tions are  consistent : 

'  5  a;  —  ?/  =  0, 
32^  =  40+ 15a?. 

17.  By  drawing  their  graphs,  show  that  Sx  —  2y  =  Q,x-\-y  =  l, 
and  4  a?  —  2/  =  13  have  a  common  solution.     What  is  it  ? 

18.  Show  by  their  graphs  that  3a;  —  2y  =  12,  4a;  —  3^  +  1  =  0, 
and  2x  —  y  =  ll  have  no  common  solution. 

19.  Determine  graphically  whether  or  not  3  a;  +  2  y  =  18, 
a;  —  y  =  8,  and  4  a;  +  2/  =  4  have  a  common  solution. 

20.  Show  graphically  that  in  general  three  or  more  linear 
equations  containing  the  same  two  unknown  numbers  cannot  have 
a  common  solution.     When  would  they  have  a  common  solution  ? 

133.  Systems  of  Fractional  Equations.  —  As  a  general  rule,  in 
solving  a  system  of  fractional  equations,  it  is  best  first  to  clear 
the  equations  of  fractions.  But  in  certain  cases,  as  when  the  un- 
known numbers  occur  only  in  monomial  denominators,  it  is  best  not 
to  clear  of  fractions  before  eliminating.  If  such  equations  were 
cleared  of  fractions,  they  usually  would  not  be  linear,  and  also 
would  give  solutions  which  would  not  satisfy  the  original  system. 

^-^=2,  (1) 

EuuPLE  1.  — Solve    \  "      " 

5  +  5  =  6.  (2) 

a     b 


SYSTEMS   OF  LINEAR   EQUATIONS 


247 


Subtracting  (1)  from  (2), 

Solving, 

Substituting  ^\  for  h  in  (1), 

Solving, 


1§  =  3. 
h 

1  =  2. 


ExAMPLK  2.  —  Solve 


3x      2y 
_3___2_^3 

^2x      Zy 


Multiplying  (1)  by  |,  to  make  the  fraction  in  x  the  same  as  in  (2), 

2a;      8y       4* 
Subtracting  (2)  from  (3),      ^7  ^  ^  ^  45  _  3^ 


Solving, 


97 


Substituting for  y  in  (1) 


198 


Solving, 


3a; 


4 

594 
194 

a;  =  ^. 


5. 


(1) 
(2) 

(3) 


EXERCISES 


Solve 


2. 


m      w 
m     w 

3     6^1 
LP     9     2* 

^  +  B      '' 


6. 


<?- 


^  +  i  =  3, 


12 
R 


20 


+  3=0. 


^  +  1  +  12  =  0, 


4 


1  =  1. 


'^ 


248 


ELEMENTARY  ALGEBRA 


«. 


LO. 


2a     56       ' 

5  +  5  =  7. 
a     0 

5t     2T       ' 

5 +  -1=29. 


11. 


12. 


4.A  2B     8' 

2^1  ^28 

9a;  2/ 

5x  2y 


Solve  for  ic  and  y : 
5 


13. 


a  , 

-  +  -  =  c, 

a;     ^ 

a;       2^ 


15. 


14. 


1, 


ma;      ny 

A_A=i 

ma;      ny 


m   .    n  _m-^n 
nx     my        mn 

i3 AVI  3 


n 
mx 


m 


w 


m" 


ny 


16.  A  water  tank  can  be  filled  by  two  pipes  in  8f  minutes.  If 
the  first  is  left  open  10  minutes  and  the  second  8  minutes,  the 
tank  will  be  filled.  In  what  time  can  each  pipe  alone  fill  the 
tank? 

17.  Two  steam  pumps  together  can  fill  a  reservoir  with  water 
in  6|-  hr.  If  one  pump  works  8  hr.  and  the  other  3  hr.  48  min., 
the  reservoir  will  be  filled.  How  long  would  it  take  each  pump 
alone  to  fill  it  ? 

18.  In  a  factory  which  operates  two  sizes  of  machines  it  is 
found  that  2  large  machines  and  5  small  ones  can  turn  out  a  cer- 
tain quantity  of  goods  in  12  hr.,  while  4  large  ones  and  3  small 
ones  can  do  it  in  10  hr.  Find  how  long  it  would  require  one 
machine  of  each  size  alone  to  turn  out  the  goods. 

19.  A  and  B  together  can  do  a  piece  of  work  in  13^  days. 
After  they  have  worked  6  days  B  leaves,  and  A  finishes  the  work 
in  16|-  days  more.  In  how  many  days  could  each  of  them  alone 
do  the  work  ? 


SYSTEMS   OF  LINEAR   EQUATIONS  249 

20.  A  current  of  electricity  flows  through  two  branches  of  a 
circuit.  The  total  resistance  to  the  current  is  found  to  be  40  ohms. 
When  the  resistance  of  one  branch  is  made  three  times  as  great, 
the  resistance  of  the  whole  circuit  is  doubled.    If  rj  and  rg  are  the 

111 

resistances  of  the  branches,  it  is  known  that   — | —  =  —  and 

?-i      j'a     40 

1 —  =  — •     Find  the  resistapice  of  each  branch. 

Sri     r^     80  T 

134.  Systems  involving  Three  Unknown  Numbers.  —  Some  prob- 
lems involving  three  unknown  quantities  may  be  solved  by  first 
expressing  them  by  means  of  a  system  of  three  linear  equations 
^containing  three  unknown  numbers.  To  solve  such  a  system  we 
/first  eliminate  one  of  the  unknown  numbers  from  any  two  of  the 
equations,  then  eliminate  the  same  number  from  one  of  these  two 
equations  and  the  third  equation.  This  will  give  rise  to  two 
new  equations  which  contain  only  two  unknown  numbers.  These 
two  equations  may  then  be  solved  as  a  new  system  by  the  methods 
of  the  preceding  sections. 

■  x  +  2y  +  2z  =  n,  (1) 

Example.  —  Solve         ■  2x+    y  +    z=    7,  (2) 

Sx  +  iy+    z  =  U.  (3) 

By  subtraction,  eliminate  x  between  (1)  and  (2). 

This  gives  3  y  +  3  2  =  15.  (4) 

"  By  subtraction,  eliminate  x  between  (1)  and  (8). 

This  gives  2  y  +  5  0  =  19.  <6) 

Eliminating  y  between  (4)  and  (5), 

z  =  S, 
By  substituting  3  for  z  in  (5)  we  get 

y  =  2. 
Now,  substituting  the  values  of  both  y  and  2;  in  (1),  we  get 

x  =  l. 

From  a  system  of  four  linear  equations  with  four  unknown 
numbers  we  can,  in  like  manner,  derive  a  new  system  of  three 
equations  with  three  unknown  numbers ;  and  so  on. 


250 


ELEMENTARY  ALGEBRA 


EXERCISES 


Solve; 


1. 


4. 


5. 


6. 


7. 


a  -f-  6  -f-  c  =  9, 
2a4-6-c  =  0, 
3a— 6  +  c  =  5. 

P-2Q+i2  =  6, 
P+3Q  +  2i2  =  13, 
2  P  -  Q  +  P  =  13. 

2^  +  2^  +  3^  =  4, 
3i^  +  4'y  +  6w  =  7, 
w  +  2'u  +  6t«  =  4. 
5  n  +  6  9-2  +  7  rg  =  3, 
10ri-12r2  +  21r3  =  3, 
15  i\  -  6  ra  +  14  ra  =  4. 
^  +  5  =  1, 

^4+  (7+6  =  0. 
2a;  +  2/-2;=2, 
a;_22/-3^  =  6, 
y-2  +  l  =  0. 
2^  +  A;  =  7  +  ?, 

l-k  =  l. 


9. 


10. 


11. 


•2p-2g  +  3r  =  10 

3p-\-q-~r=5, 

p  —  q-j-2r  =  7. 

IHH=''' 

10     10      6       ' 

a     b      c       - 
4  +  5-i5  =  ^- 

«     y 

[y    ^ 

^l+^i'o  =  ^' 

1    .    4  _1 
wl     5i>      12' 

17        3 
[p'^30     4.71 

12.  The  perimeter  of  a  triangle  is  28  in.  Two  of  the  sides  are 
3qual,  and  their  sum  exceeds  the  third  side  by  4  in.  Find  the 
lengths  of  the  sides. 

13.  The  sum  of  the  two  sides  of  a  triangle  which  meet  at  one 
vertex  is  44  in.,  at  another  vertex  40  in.,  and  at  the  third  vertex 
36  in.     Find  the  lengths  of  the  three  sides. 

14.  The  sum  of  the  three  angles  of  any  triangle  is  180°.  If 
two  angles  of  a  triangle  are  equal,  and  each  is  5°  more  than  the 
third,  find  the  size  of  each. 


SYSTEMS   OF  LINEAR  EQUATIONS  251 

15.  In  triangle  PQR,  angle  P  is  16°  less  than  angle  Q,  and 
angle  Q  is  4°  less  than  angle  B.     How  man}'  degrees  in  each  ? 

16.  It  is  known  in  geometry  that  two  tangents   drawn  from 
the  same  point  to  a  circle  are  equal.     Hence,  if 
a  circle  is  inscribed  in  a  triangle,  as  in  the  figure, 
the  points  of  contact  divide  the  sides  of  the  tri- 
angle into  three  sets  of  equal  tangents. 

If  AB=12  in.,  5(7=16  in.,  and  AO=U  in., 
show  that  x  +  y  =  14,  a;  +  2;  =  16,  and  y-{-z  =  12. 
Find  the  distances  of  the  points  of  Contact  of  the  circle  with  the 
three  sides  from  the  vertices. 

17.  The  sides  of  a  triangle  are  24  in.,  32  in.,  and  38  in.  Find 
the  points  where  the  inscribed  circle  touches  the  sides. 

18.  Three  circles  are  to  be  drawn  tangent  to  each  other,  and 
with  their  centers  at  three  given  points  A,  B,  and 
C,  respectively.  If  the  distance  between  A  and 
5  is  10  in.,  between  B  and  C  13  in.,  and  between 
A  and  C  15  in.,  find  the  radii  of  the  circles. 

19.  In  accurate  tool  work  where  holes  are  to 
be  bored  close  together  in  a  metal  plate,  the  centers 
of  the  holes  are  first  marked  carefully  to  thou- 
sandths of  an  inch.  This  may  be  done  by  first  turning  out  disks 
on  a  lathe  of  such  sizes  that  when  placed  tangent  to  each  other 
their  centers  mark  the  positions  of  the  centers  of  the  required 
holes.  These  disks  are  then  fastened  on  the  metal  plate  in 
tangent  positions,  and  the  holes  bored  at  their  centers. 

Three  holes  are  to  be  bored,  the  distances  between  whose 
centers  shall  be  0.^60  in.,  0.620  in.,  and  0.844  in.,  respectively. 
Find  the  radii  of  the  required  disks.     See  Problem  18. 

20.  A  certain  number  consists  of  three  digits  whose  sum  is  9. 
If  198  be  subtracted  from  the  number,  the  remainder  will  consist 
of  the  same  digits  in  a  reverse  order ;  and  if  the  number  be  di- 
vided by  the  hundreds'  digit,  the  quotient  will  be  108.  What  is 
the  number  ? 


252 


ELEMENTARY  ALGEBRA 


21.  It  is  given  that  the  standard  daily  ration  for  an  adult 
laboring  man  should  include  16  oz.  starch,  4  oz.  fat,  and  4  oz. 
albumen.  The  amount  of  these  materials  in  bread,  butter,  and 
beef  are  as  follows : 


Food 

Starch 

Fat 

Albumen 

Bread 
Butter 
Beef 

54% 
0% 
0% 

1% 
83% 

15% 

9% 
1% 

15% 

Find  the   quantities   of  these   foods  required  to  make  a  daily- 
ration. 

22.  Potatoes  contain  20  %  starch,  1  %  fat,  2  %  albumen  ;  pork 
contains  no  starch,  26  %  fat,  13  %  albumen ;  and  cream  contains 
4  %  starch,  18  %  fat,  2  %  albumen.  Find  the  quantities  of  these 
foods  required  to  make  a  standard  daily  ration.  '  "^ 

23.  There  are  three  compounds  composed  of  different  metals. 
The  first  contains  7  parts  (in  weight)  silver,  3  parts  copper,  and 
6  parts  tin;  the  second  contains  12  parts  silver,  3  parts  copper, 
aud  1  part  tin ;  the  third  contains  4  parts  silver,  7  parts  copper, 
and  5  parts  tin.  How  much  of  each  of  these  three  compounds 
must  be  taken  in  order  to  form  a  fourth  which  shall  contain  8  oz. 
of  silver,  3f  oz.  of  copper,  and  4^  oz.  of  tin  ? 

24.  In  an  athletic  contest  team  A  won  with  a  total  of  35 
points,  team  B  got  second  place  with  a  total  of  32  points,  and 
team  C  got  third  place  with  a  total  of  31  points,  as  shown  in  the 
table : 


Team 

ISTS 

2d8 

8d8 

Total 

A 

7 

3 

1 

35 

B 

6 

4 

4 

32 

C 

2 

7 

9 

31 

How  many  points  does  each  place  count  ? 


SYSTEMS   OF  LINEAR   EQUATIONS 


253 


SUPPLEMENTARY  EXERCISES 


Solve : 


1. 


2. 


5M,N_^ 

2M     N     o 

4  TF+w^2 
6  Tr+  w     5' 

^  +  ^  =  ^- 
ri  +  r,  +  l       ' 
rj  +  l      ra-l' 


Solve  for  x  and  y : 


a_6_ 
3a;     4y 


Solve: 

a+6  +  c  =  2, 
6  +  c  +  d  =  4, 
c  H-  c?  +  «  =  6, 
la  +  6  +  d  =  3. 
r  +  s  4-  ^  =  18, 
r  +  s  + 10  =  18, 
s  +  w;  +  « =  18, 
r+w>+^=18. 


9. 


10 


1    +^=0. 


a+6     a— 6 
a-36 


h-2a 


2. 


5. 


6. 


^   1-^=1. 


l  +  a;      1  +  2/ 

_2 1_^1 

1  +  a;     1  +  ^     2 


a:-2     y^l 
1      .      6 


1, 


.aj-2     y  +  1     8 


r3w  .  w-1 


8. 


a?  2/ 

i2x         y 


=  w. 


=  «. 


11. 


12. 


«  +  2/+24-««  =  10, 
a;  —  y  —  2  +  w  =  0, 
2x  +  y-\-Sz  —  w  =  % 
^x  —  y+z-2w  +  ^  =  0. 

2p-q-r  +  s=lZ, 
p  +  q  —  r  —  s-\-l=zOj 

3p-f-2g-r  +  2s  =  17. 

13.  Find  a  number  of  three  digits,  such  that  the  sum  of  the 
digits  shall  be  15;  the  sum  of  the  hundreds'  digit  and  the  ones' 
digit  shall  be  1  less  than  the  tens'  digit ;  and  the  ones'  digit  sub- 
tracted from  4  times  the  hundreds'  digit  shall  equal  the  tens'  digit 


254  ELEMENTARY  ALGEBRA 

14.  A  merchant  has  two  kinds  of  tea.  If  he  mix  3  pounds  of 
the  poorer  with  7  pounds  of  the  better,  the  mixture  will  be  worth 
76i^  a  pound;  but  if  he  mix  7  pounds  of  the  poorer  with  3  pounds 
of  the  better,  the  mixture  will  be  worth  68|-^  a  pound.  What  is 
the  price  of  each  kind  of  tea  ? 

15.  There  are  two  alloys  of  copper  and  silver,  of  which  one  con- 
tains 3  times  as  much  copper  as  silver,  and  the  other  contains  5 
times  as  much  silver  as  copper.  How  much  must  be  taken  of 
each  alloy  to  make  7  pounds,  of  which  half  shall  be  silver  and  the 
other  half  copper  ? 

16.  If  the  altitude  of  a  rectangle  be  increased  4  inches,  and  its 
base  diminished  2  inches,  the  area  will  be  increased  22  square 
inches ;  and  if  the  altitude  be  increased  1  inch,  and  the  base  dimin- 
ished 1  inch,  the  area  will  be  increased  2  square  inches.  Find 
the  base  and  the  altitude  of  the  rectangle. 

17.  A  company  of  men  rented  a  yacht.  When  they  paid  their 
rental  they  found  that  if  there  had  been  2  more  persons  to  pay 
the  same  bill,  each  would  have  paid  50  cents  less  than  he  did ; 
and  if  there  had  been  2  fewer  persons,  each  would  have  paid  $  1 
more  than  he  did.  Find  the  number  of  persons  and  the  amount 
that  each  paid. 


CHAPTER   XIII 
SQUARE  ROOT.     QUADRATIC   SURDS 

135.  Square  Roots  of  Polynomials. — Since  (a+6)^=a^-|-2  a6+6', 


Note.  —  As  shown  in  §  72,  every  quantity  has  two  square  roots,  differing 
only  in  sign.  Thus,  the  other  square  root  of  a^  -\-  2  ab  -\-  b^  is  —  (a  +  6), 
or  —  a  —  6.  In  this  chapter  we  shall  consider  only  one  of  the  square  roots 
of  any  expression,  viz.  the  one  of  which  the  first  term  is  positive. 

A  study  of  the  above  identity  will  reveal  the  process  of  finding 
the  square  root  of  any  polynomial  which  is  a  perfect  square. 

If  the  polynomial  has  three  terms  arranged  according  to  the 
powers  of  one  letter,  these  correspond  to  the  terms  of  a^+2  ab-\-bK 
Evidently,  the  first  term  a  of  the  root  is  a  square  root  of  al  If 
a^  be  subtracted  from  the  trinomial,  the  remainder  is  2ab-\-  b^ 
The  second  term  of  the  root,  b,  may  be  found  by  dividing  2  a6, 
the  first  term  of  the  remainder,  by  2  a,  or  twice  the  term  of  the 
root  already  found. 

The  work  is  usually  arranged  as  follows : 

a^-\-2ab  +  b^\a-{-b 

/,2 


2a 

2a  +  6 


2ab  +  b^ 
2ab-\-b^ 


The  divisor  2  a,  used  in  finding  the  second  term,  is  called  the 
trial  divisor.  AVhen  the  second  term  b  is  added  to  the  trial  divisor, 
it  gives  the  true  divisor,  2  a  +  6,  because  when  the  latter  is  multi- 
plied by  b  it  gives  the  entire  remainder  2  ab-\-  b\ 

255 


256  ELEMENTARY  ALGEBRA 


Example  1.  — Find  the  square  root  of  Oic^  _}.  4  y'i  __  12  xy. 

Writing  the  expression  in  descending  powers  of  a:,  the  work  is  as  follows 

9  a;2  -  12  icy  +  4  y2|3  x-2y 

Q'»-.2 

—  12  icy  +  4  y^ 


6x 
6x-2y 


—  12xy  +  4:y^ 


The  first  term  of  the  root  is  Vdx^,  or  Sx.  Subtracting  9a;2  leaves 
— 12  xy  -f  4  y^.  The  trial  divisor  is  2(3  a;),  or  6  x.  Dividing  —  12xy  hj  dx 
gives  —  2«/,  the  second  term  of  the  root.  Adding  —2y  to  6a;,  the  trial 
divisor,  gives  the  true  divisor  6x  —  2y.  Multiplying  this  by  —  2  y  gives 
—  12  a;y  +  4  2/2  exactly,  which  shows  that  6  x  —  2  y  is  the  entire  root. 

If  the  square  root  of  a  polynomial  has  three  or  more  terms,  the 
first  two  may  be  found  as  above ;  then  by  grouping  terms,  these 
two  may  be  used  as  one,  and  the  third  term  obtained  by  a  repeti- 
tion of  the  process  used  to  obtain  the  second.  Similarly,  by 
grouping  the  first  three  terms  of  the  root,  the  fourth  may  be 
found,  etc. 

Example  2.  —  Find  the  square  root  of  8  w  —  4  w^  +  w*  +  4. 
First,  arrange  the  terms  in  descending  powers  of  n. 

4w3  +8n  +  4|w2-2w-~2 


n* 

2/l2 

2n2-2w 

-4n8              +8w  +  4 
-4n3  +4n2 

2w2-4  7i 
2  ?i2  _  4  ^ 

-2 

-4w2  +  8w  +  4 
-  4  w2  _|_  8  n  4-  4 

The  first  term  of  the  root  is  \/#,  or  n^.  Hence  the  Jirst  trial  divisor  is 
2  w2.  Dividing  —  4  n^,  first  term  of  the  remainder,  by  2  n^  gives  ~  2n, 
second  term  of  the  root.  Adding  this  to  2  n^  gives  2  n^  —  2  n,  the  first 
true  divisor.  Multiplying  this  by  —  2n  gives  —  4  w^  +  4  n2.  Subtracting 
this  product  from  the  first  remainder  leaves  —  4n2  +  8  w  +  4,  the  second 
remainder.  Now  using  the  root  found,  n^—2  n,  as  one  term,  the  second 
trial  divisor  becomes  2(^2  _  2  w)  or  2^2  — 4  n.  Dividing  the  first  term  of 
this  into  the  first  term  of  the  remainder  gives  —  2,  the  third  term  of  the  root. 
Adding  —  2  to  the  trial  divisor  gives  2  »2  _  4  ^  _  2,  the  second  true  divisor. 
This  multiplied  by  —  2  gives  the  second  remainder,  which  shows  that 
w2  —  2  w  —  2  is  the  entire  root. 


SQUARE  ROOT.     QUADRATIC  SURDS  257 

Note.  — Care  must  be  taken  to  first  arrange  the  terms  in  any  problem  in 
descending  or  ascending  powers  of  some  letter.  Each  remainder  and  each 
divisor  must  also  be  arranged  like  the  original  expression. 

The  above  examples  illustrate  the  general  rule: 

(1)  Write  the  given  polynomial  in  descending  or  ascending  powers 
of  some  letter. 

(2)  Take  the  square  root  of  the  first  terin  for  the  first  term  of  the 
root,  and  subtract  the  first  term,  from  the  polynomial. 

(3)  Double  the  root  found  for  the  first  trial  divisor,  divide  the  first 
term  of  the  remainder  by  this,  and  write  the  quotient  as  the  second 
term  of  the  root. 

(4)  Add  this  quotient  to  the  trial  divisor  to  obtain  the  true  divisor, 
multiply  the  true  divisor  by  the  second  term  of  the  root,  and  subtract 
the  product  from  the  preceding  remainder. 

(5)  If  there  is  still  a  remainder,  double  all  of  the  root  already 
found,  for  a  new  trial  divisor,  and  proceed  as  before.  Continue  this 
process  until  all  terms  of  the  root  are  found, 

EXERCISES 

Find  the  square  roots  of  the  following : 

1.  4a2  +  20a  +  25. 

2.  l-16y4-642/». 

3.  m*  +  25  w^- 10  m^n. 

4.  5ar'  +  a;*-2a^-|-4-4a;. 

5.  4<^  +  49-3^-70«  +  20^. 
Q.  l-^2b-b^  +  :^W-2b^  +  b\ 

7.  49  ^«  +  42^« -19^^-12 ^2 _|.4^ 

8.  r*  +  21r2  +  4-10r34-20r. 

9.  a*  +  4a36  +  6a2&2  +  4a63+6*. 

10.  4w;^-4w;''  +  17?c--8z«4-16. 

11.  l^M^N^-\-M^-^M^N-\2MN^-\-4.N*. 

12.  a;^  -  2  a^?/  -  2  a;/  4-  3  ^y"^  +  y\ 

13.  9i)<  -  12 p^q  -  26i)¥  _^  20p<f  +  25  q^ 


258  ELEMENTARY  ALGEBRA 

14.  49-42  F+37F2-}-4F''-12  F^. 

15.  a^-2a;«  +  3a;*-4a^H-3«2_2a;_|_i. 

.   16.   m«— 6m^/i4-15mV- 20m%«  +  15mV  — 6m/i*  +  n«. 
17.   -y'  + 16  w;^  -  4  v^w  + 10  v^w"^  -  24  vw'  -^  20  vV  +  25  2;W. 
•    18.   4P«-20P^  +  46P2^.4ip4^9_52P3-24P. 

136.  Square  Roots  of  Arithmetical  Numbers.  —  Any  arithmetical 
number  of  two  or  more  figures  is  in  nature  a  polynomial. 

Thus,  6263  =  5000  +200  +  60+3 

=  5  X  103  +  2  X  102  +  6  X  1&  +  3. 

Hence,  the  square  root  of  an  arithmetical  number  may  be 
obtained  in  practically  the  same  manner  as  that  of  a  polynomial. 

The  square  root  of  1  is  1;  of  100  is  10;  of  10000  is  100;  etc. 
Hence  the  square  root  of  a  number  between  1  ar^d  100  is  between 
1  and  10;  of  a  number  between  100  and  10000  it  is  between  10 
and  100 ;  etc.  That  is,  the  integral  part  of  the  square  root  of  an 
integer  of  one  or  two  figures  contains  one  figure ;  of  an  integer  of 
three  or  four  figures  it  contains  two  figures  ;  etc.     Therefore, 

If  the  figures  of  an  integer  are  marked  off  from  right  to  left  into 
groups  of  two,  the  number  of  figures  in  the  integral  part  of  the  square 
root  will  he  equal  to  the  number  of  groups,  any  one  figure  that  re- 
mains on  the  left  being  counted  as  a  whole  group. 

Thus,  marked  off  into  groups,  20684  will  become  2'06'84'.  Since  there  are 
three  groups,  the  integral  part  of  the  square  root  will  contain  three  figures. 

If  the  square  root  of  a  number  be  a  number  of  two  figures,  the 
tens  of  the  root  may  be  denoted  by  a  and  the  ones  by  b.  Then 
the  root  will  be  expressed  by  a  +  6,  and  hence  the  number  by 
a^  +  2 a6  +  b"^.  The  use  of  the  fact  that  V(?+2a6T^  =  a -/- 6  is 
best  shown  by  examples. 

Example  1.  — Find  the  square  root  of  3969. 

Pointing  off,  we  have  o9'69'.  Since  tliere  are  two  groups,  there  must  be 
two  figures  in  the  root.  The  root  lies  between  60  and  70,  because  60^  is  less 
than  3969,  and  702  jg  greater  than  3969.  That  is,  the  tens'  figure  of  the  root 
is  6,  the  square  root  of  the  largest  square  in  the  left-hand  group  of  the  given 
number. 


SQUARE  ROOT.     QUADRATIC  SURDS  259 

The  work  is  indicated  thus  : 

I  a  +  b 
39^69160  +  3=63 
a2  =         36  00 
2  a  =  120 
2  a  +  6  =  123 


3  69 
3  69 


Since  a  =  60,  a^  =  3600.  Subtracting,  the  remainder  is  369.  The  trial 
divisor  2  a  becomes  2  x  60,  or  120.  Dividing  369  by  120  gives  approximately 
3.  Hence  b  is  probably  3.  Hence  the  trial  divisor  2  a  -\-  b  becomes  123. 
Multiplying  123  by  3  gives  369.  Taking  this  from  the  first  remainder  leaves 
zero.     Hence,  60  +  3,  or  63,  is  the  exact  root. 

Example  2.  —  Find  the  square  root  of  203401. 

By  pointing  off  it  is  seen  that  there  are  three  figures  in  the  root. 

a  +  6  +  c 
20'34'01'  1400  +  50  +  1  =451 
a2  =  16  00  00 


2a  =  800 

2  a  +  6  =  850 

2  (a  +  &)  =  900 

2(a  +  6)+c  =  901 


4  34  01 
4  25  00 


9  01 
9  01 


The  largest  square  in  20  is  16.  Hence  a^  =  160000,  and  a  =  400.  The 
first  remainder  is  43401.  The  first  trial  divisor  2  a  is  2  x  400,  or  800.  Divid- 
ing 43401  by  800  gives  approximately  50,  the  value  of  6.  The  first  true  divisor 
is  800  +  50,  or  850.  Multiplying  850  by  50  gives  42600.  Subtracting,  the 
second  remainder  is  901.  The  second  trial  divisor  2(a  +  &)  is  2  x  450,  or 
900.  901  divided  by  900  gives  approximately  1,  the  value  of  c.  The  second 
true  divisor  is  900  +  1,  or  901.  Multiplying  901  by  1  gives  901.  Subtract- 
ing this  from  the  second  remainder  leaves  zero.  Hence  the  exact  root  is 
400  +50  +  1,  or  451. 

When  the  square  root  of  a  number  has  decimal  places,  the 
number  itself  has  twice  as  many. 

Thus,  0.232  =  0.0529. 

Hence,  to  mark  off  a  number  which  contains  a  decimal,  begin  at 
the  decimal  point  and  mark  to  the  left  and  to  the  right,  jnUting  two 
figures  in  each  group. 

If  a  decimal  has  an  odd  number  of  figures,  add  a  zero  to  make 
it  even. 

Thus,  723.618  will  become  7  23'.61'80'. 


260 


ELEMENTARY  ALGEBRA 


Example  3.  — Find  the  square  root  of  50.9796. 


60.  '97  '96'  1 7.00  +  .10  +  .04  =  7.14 
49.  00  00 


14.00 
14.10 
14.20 
14.24 


1.  97  96 
1.  41  00 


.  56  96 
.56  96 


An  approximation  to  the  value  of  the  square  root  of  a  number 
that  is  not  a  perfect  square  may  be  found  to  any  degree  of  accu- 
racy desired. 

Example  4.  —  Find  to  three  decimal  places  the  square  root  of  2. 
Since  three  decimal  places  are  desired  in  the  root,  we  annex  6  zeros. 

2.  'OO'OO'OO'  1 1  +  .4  +  .01  +  -004  =  1.414 

1.  00  00  00 


2 
2.4 

1.00  00  00 
.  96  00  00 

2.8 
2.81 

.  04  00  00 
.  02  81  00 

2.82 
2.824 

.  01  19  00 
.  01  12  96 

EXERCISES 

rind  the  square  root  of : 

1.  7396.              4.   26244.  7.  16129.  10.  146.41. 

2.  1849.              5.   41209.  8.  96.4324.  11.  125.44. 

3.  5776.               6.   17424.  9.  12.25.  12.  4.6225. 

Find  to  three  decimal  places  the  square  root  of: 

13.  5.  15.   0.6.  17.   28.  •        19.    37.1. 

14.  3.  16.    1.2.  18.    0.45.  20.    0.75. 

21.  The  area  of  a  triangle  whose  sides  are  a,  h,  and  c  equals 
Vs(s  —  a){s—  h){s  —  c),  where  s  equals  one  half  of  the  sum  of  the 
three  sides.  The  sides  of  a  triangular  field  are  42  rd.,  38  rd.,  and 
46  rd.,  respectively.  Find  its  area  to  hundredths  of  a  square 
rod. 


SQUARE  ROOT.     QUADRATIC  SURDS  261 

Note.  — The  use  of  square  root  in  solving  equations  will  be  shown  in  the 
next  chapter. 

137.  Real  and  Imaginary  Numbers.  —  Since  any  even  power  of 
either  a  positive  or  a  negative  number  is  positive,  any  even  root 
of  a  negative  number  is  neither  a  positive  nor  a  negative  number. 
The  indicated  even  root  of  a  negative  number  is  called  an 
imaginary  number. 

Thus,  V— 4,  \/—  9,  v'—  1,  etc.,  are  imaginary  numbers. 

A  number  that  is  not  imaginary  is  called  a  real  number.  Hence, 
all  numbers  are  classified  as  either  real  or  imaginary.  All 
numbers  with  which  we  have  had  to  deal  up  to  the  present  time 
are  real  numbers. 

NoTB.  —  In  the  chapter  following  this  some  imaginary  numbers  will  be 
encountered  in  solving  equations.  But  no  knowledge  of  the  operations  with 
these  numbers  will  be  needed  in  the  First  Course.  A  thorough  treatment  of 
imaginary  numbers  will  be  found  in  the  Second  Course. 

138.  Surds.  —  An  indicated  root  which  is  real  but  the  value  of 
which  cannot  be  computed  exactly  is  called  a  surd. 

Thus,  V3,  VS,  y/m^\^,  etc. ,  are  surds. 

A  surd  whose  index  is  2  is  called  a  quadratic  surd. 

Note.  —  In  this  chapter  are  given  only  a  few  of  the  important  processes 
applied  to  quadratic  surds.  A  fuller  treatment  of  surds  will  be  found  in  the 
Second  Course. 

Every  positive  number  has  two  square  roots,  differing  only  in 
sign.  But  in  dealing  with  quadratic  surds  only  the  positive  value 
of  each  will  be  considered. 

139.  Reducing  a  Surd  to  its  Simplest  Form.  —  A  quadratic  surd 
is  in  its  simplest  form  when  the  expression  under  the  radical  sign 
is  integral  and  contains  no  factor  of  which  the  indicated  root  can 
be  found.  A  quadratic  surd  that  is  not  in  its  simplest  form  can 
be  reduced  to  its  simplest  form  by  use  of  the  principle  that 


262  ELEMENTARY  ALGEBRA 

The  square  root  of  the  product  of  two  numbers  equals  the  product 
of  the  square  roots  of  the  numbers. 

Thus,  V30  =  Vi  X  V9,  because  each  equals  6. 

■\/25  v'^iv  -  wy  =  V25v^  X  y/{v-  to)*,  because  each  equals  5  v(v  —  w)^. 

The  application  of  this  principle  in  simplifying  surds  is  shown 
in  the  following  examples : 

Example  1.  —  Reduce  to  its  simplest  form  V63. 

Factoring  63  into  two  factors  one  of  which  is  the  largest  perfect  square 
possible,  we  have 

VeS  =  v'Ox^  =  V9  X  V?  =  3V7. 

Example  2.  —  Reduce  to  its  simplest  form  \/|. 

Multiplying  both  terms  of  f  by  5  to  make  the  denominator  a  perfect 
square,  then  factoring,  we  get 


Example  3.  —  Reduce  to  its  simplest  form  \/32  a'^b^c^. 

y/W^b^  =  Vie  a^^d^  x  2  c  =  VWa^b^  X  V2c  =  4  ab^cV2c. 

If  the  expression  under  the  radical  sign  is  integral,  separate  it  into 
two  factors  of  which  one  is  the  largest  perfect  square  possible.  The 
square  root  of  the  square  factor  becomes  the  coefficient  of  the  radical. 

If  the  expression  under  the  radical  sign  is  a  fraction,  multiply  both 
terms  of  the  fraction  by  such  a  quantity  as  will  make  the  resulting 
denominator  a  perfect  square.  Then  separate  into  two  factors  and 
proceed  as  above. 

Note.  — The  importance  of  knowing  how  to  simplify  a  quadratic  surd  is 
seen  when  the  approximate  value  of  a  surd  is  to  be  found.  Thus,  if  one  knows 
that  V2  =  1.414,  then  \/200  =  10  v^  =  14.14.  Also  if  it  is  known  that  VS  = 
1.732,  then  V27  =  3\/3  =  5.196. 


EXERCISES 

Reduce  to  the  simplest  form : 

1.    V8.            4.    V27.        7.    V28. 

10. 

V54. 

13.    V32. 

2.    Vi2.          5.    Vl8.        8.    V48. 

11. 

V72. 

14.    V80. 

3.    V24.          6.    V50.        9.    V75. 

12. 

Vl25. 

15.    V242, 

SQUARE   ROOT.     QUADRATIC  SURDS                263 

22.  V^.         25.  V}f         28.    VS- 

23.  VH.         26.  Vfj.         29.    V^. 

24.  V^.         27.  V^.         30.    VJi- 
35.    V8  fiT\  39.    V45AB^. 


36.    V50P^^.  40.    -^a\a^-b)\ 

33.  V25  m»ii.  37.    VWWy,  c^ 

34.  V49 it^y.  38.    VSfcV.  '    ^fcd^* 

42.  If  V2  =  1.414,  find  the  values  of  the  surds  in  Exercises  1, 
5,  6, 11, 13, 15, 16,  and  19. 

43.  If  V3  =  1.732,  find  the  values  of  the  surds  in  Exercises  2, 
4,  8,  and  9. 

140.  Addition  and  Subtraction  of  Surds.  —  Only  quadratic  surds 
that  have  the  same  number  under  the  radical  sign  can  be  added 
or  subtracted.  They  are  added  or  subtracted  just  as  any  two 
similar  terms. 

Thus,  just  as3n  +  2n  =  5n,    so  3\/7  +  2\/7  =  5V7. 

It  usually  is  best  first  to  reduce  surds  to  be  added  or  subtracted 
to  their  simplest  forms  before  trying  to  unite  them. 

Thus,  Vl2  +  v^  +  V75  =  2  V3  +  3V3  +  5  V3  =  10V3. 

To  add  or  subtract  quadratic  surds  having  the  same  radical  part, 
add  or  subtract  the  coefficients  of  the  radical  part, 

EXERCISES 

Simplify : 

1.  V2+V8.  4.    V45-V20.  7.    2V2-}-V32. 

2.  V3  +  V12.  5.    V32-V8.  8.    5V5-V45. 

3.  V18  +  V27.  6.    V50-V18.  9.   2V63-V28; 

10.  V99  +  V44.  12.    V75-~V3-Vi2. 

11.  VI8-V8+V32.  13.    V20  4-V45-V80. 


264  ELEMENTARY  ALGEBRA 

14.  V6+V294  +  V24.  22.    V^'-S^/V^. 

15.  2V2  +  3V18-V50.  23.    ^/9a^  +  o  abVa^b. 


16.  V|  +  3V-|.  24.  Vmw^  +  ri V 9  mn. 

17.  3V48-V|-2V|.  25.  8V2^-'yV50v¥. 

18.  Vf4-5Vii^-2V|.  26.  18pV3^  +  2gV27^. 

19.  V20  4-3V5  +  Vi.  27.  rV^  +  V^  +  V^ 


20.    Vi-  —  4 V3  —  6 Vl«  28.    -Vu^vw  +  ^uv^w  +  ^uvw\ 


21.    2aVa^4-3Va^.  29.    Va^ftc  +  Vo^d^c  +  Va^^c^ 

141.  Multiplication  of  Surds.  —  The  product  of  two  quadratic 
surds  is  obtained  by  use  of  the  principle  in  §  139,  i.e. 

The  product  of  the  square  roots  of  two  numbers  equals  the  square 
root  of  the  product  of  the  numbers. 

Thus,   \/2  X  \/3  =  V2l<~3  or  V6  ;  VT5  x  VB  =  VIS  x  5  =  VTS  =  5  V3. 
Similarly,      Va  x  V&  =  y/ah  ;  Vm  x  Vmn  =  Vrn'-^w  =  my/n. 

The  product  of  two  expressions  involving  surds  is  obtained  just 
as  the  product  of  any  two  polynomials,  i.e.  by  multiplying  each 
term  of  one  by  each  term  of  the  other,  and  adding  the  similar 
terms  obtained. 

Example.  — Multiply  3  +  V2  by  2  -  \/2. 
3+V2 
2-V2 


6  +  2V2 

-3\/2-2 

.V2 

6-    V2-2=4- 

EXERCISES 

Multiply : 

1.    V3  by  V5. 

5.    V2  by  V6. 

9. 

V2xvn. 

2.    V2  by  V7. 

6.    V3  by  Vl5. 

10. 

V5  X  V35  X  V2i. 

3.    V3byVn. 

7.    V2I  by  V3. 

11. 

V3  X  a/6  X  V2. 

4.    V5  by  Vl3. 

8.    ViO  by  V30. 

12. 

V8  X  V22  X  V33. 

SQUARE  ROOT.     QUADRATIC   SURDS  265 

13.  2  -  V5  by  4  +  V5.  25.  2  V7  +  3  V2  by  3  V7  -  5  Vi 

14.  1+V3by3+V3.  26.  1  -  V2  by  2V3^-3^^2. 

15.  7  -  V6  by  2  -  V6.  27.  Vr  by  Vrs. 

16.  9  -}- Vn  by  3  -  VH-  28.  aV6  by  bVa. 

17.  2  +  V3  by  3  +  V2.  29.  2  a; V^  by  a^^V^. 

18.  Vl5  — 4byV5  +  l.  30.  p^Vp  hy  pqVpq. 

19.  V7  +  1  by  3  -  V7.  31.  5  wj Vmw  by  2  v V2~m;. 

20.  Vl2  +  2  by  V3-  5.  32.  VSmii  by  VTwti. 

21.  V'3H-V2by  V'3-V2.  33.  x-^Vyhy  x-Vy. 

22.  V5  +  V6  by  V5-fV6.  34.  r  — Vsbyr  — Vs. 

23.  V7-2V2by  V7-V2.  35.  2  a -3V6  by  a+2V6. 

24.  V3+4Vnby  V2  +  VIi.  36.  Vm  +  Vj*  by  Vm  -  Vw. 

142.  Division  of  Surds.  —  In  the  division  of  expressions  involv- 
ing surds  it  is  best  to  first  free  the  divisor  of  surds.  This  is 
accomplished  by  use  of  the  principle  that  if  the  dividend  and 
divisor  are  both  multiplied  by  the  same  expression,  the  value  of 
the  quotient  is  unchanged. 

Example  1.  —  Divide  Vs  by  V2. 

Multiplying  both  dividend  and  divisor  by  \/2, 

\/3  ^  VS  X  V2  ^  V6  ^  1  ^/g 

V2      V2xV2       2        2 

If  the  divisor  is  a  binomial,  in  order  to  free  it  of  surds  ii  luust 
be  multiplied  by  the  same  binomial  with  the  sign  between  the 
terms  changed.  This  gives  the  product  of  the  sum  and  difference 
of  two  numbers,  which  is  the  difference  between  theii  squares. 

Example  2. — Divide  V7+2V5by  V7  +  Vs. 
Multiplying  both  dividend  and  divisor  by  V?  —  \/5, 

V?  +  2a/5  ^  (  V7  +  2  V5)(  V7  -  \/6)  ^  V35  -  P,  ^       r^y  _^  ^ 

VT  +  VS        (V7+  \/5)(V7-  VS^)  2 


266  ELEMENTARY  ALGEBRA 

EXERCISES 

Perform  the  indicated  divisions ; 

1.  JL.  4    V6  7.   _8_.  10.  -5L.  13.  _4^, 

V2  ■   VS'  V6  Va  V2p 

2.  J,.  g     V6  8.    -12^.  11.    V|_  14.   ^ 

V3  ■   .^2'  '^1^  Vm  V6u 

21.  — ^-i ^.  26    2V5-3V3 

svg  +  VS" 

22.  ^^^+1.  27    4V7  +  3V2_ 
'  3V7-rV2' 

1 


16. 
17. 
1  ft 

2 

2  +  V2 
4 

V3-1 
6 

1  Q 

V3  H-V2 
16 

on 

V5- V3 
12 

V7-V6  25. 


8.      ""   . 

V'18 

9    ^^ 

V8 

4 

V2I- V19 

V3  4-1 

V3-1 

V5  +  2 

V5-2 

VII- V7 

VIi  +  V7 

V6  +  2V2 

23.  _         .  28. 

V5-2 

24.  V^^VT  29. 


30. 


V6  +  V2 


X 

-\/x—  V3/ 

m  —  w 
Vm  —  V^i 


143.  Graphs  of  Quadratic  Surds.  —  Although  the  exact  value  of 
a  quadratic  surd  cannot  be  computed  decimally,  its  value  can  be 
represented  exactly  by  the  length  of  a  line.  The  principle  in- 
volved is  that  the  square  of  the  hypotenuse  of  any  right  triangle 
equals  the  sum  of  the  squares  of  the  other  two  sides. 

Thus,  if  a  right  triangle  ABC  be  drawn  with  AB  equal  to  1  and  BC  equal 
to  1,  then  AC  will  equal  \/2.  Now,  if  DCbQ  drawn  from  C,  perpendicular 
to  CA^  and  equal  to  1,  the  side  DA  of  the  right  triangle  ACD  will  equal  V3. 

By  constructing  another  right  triangle  with  AD  as  a  side  and  the  new 
side  equal  to  1,  and  continuing  this  process,  the  hypotenuse  of  a  right  tri- 
angle can  be  obtained  that  will  represent  any  quadratic  surd. 


SQUARE   ROOT.     QUADRATIC  SURDS  267 

Many  quadratic  surds  can  be  represented  by  drawing  a  single  triangle. 
Thus,  '\iAB  =  Z  and  BG  =\,AC=  VlO.    UAB  =  ^  and  BC  =  2,AC=  Vl3. 

EXERCISES 

By  drawing  right  triangles,  represent  the  following : 

1.  V5.  4.    Vs.  7.    V34.  10.    Vil. 

2.  V6.  5.    VlT.  8.    V53.  11.    ViO. 

3.  V7.  6.    Vi5.  9.    V45.  12.    V65. 

SUPPLEMENTARY  EXERCISES 

Find  the  square  root  of : 

1.  2^  +  42/»  +  42/'4-2  2/  +  4+i. 

y 

2.  M^  +  SR  +  12-^  +  ^. 

3.  2+^V2a6  +  ^^  +  — +  -3. 

^  2         ,  11  «2        3  ^         9  ^4 

4.  nr  —  tn -. 

4    ^2n^4n* 

Find  the  square  root  to  hundredths  of: 

5.  f 

Suggestion.  —  In  finding  the  square  root  of  a  common  fraction  it  is  best 
to  first  change  the  fraction  to  a  decimal.  Since  we  want  two  places  of  deci- 
mals in  the  root,  write  the  first  four  decimals  of  the  number.  Thus, 
^  =  0.7142. 

6.    f  7.    f  8.    -H-  9-    i-  10-    i- 

11.  A  method  used  by  the  Arabs,  and  as  late  as  the  Middle 
Ages,  of  finding  the  approximate  value  of  the  square  root  of  a 
number  was  to  substitute  in  the  formulae  of  approximation : 

(a)  V^N^=n  +  ^,    (b)  V^?T^=rH-— ^- 


268  ELEMENTARY  ALGEBRA 

The  true  value  of  the  root  was  found  to  be  between  the  two 
values  obtained.  For  example,  to  find  V81J,  separate  82  into 
81  -f  1.     Then  from 

(a)  V82=:V81  +  1=V92TPT  =  9  +  tV  =  ^T8;  ^^om 
(6)   V82  =  V8rTl=ViF+l  =  94-TV  =  9J9. 

Find  the  decimal  values  of  these  residts,  which  are  the  same 
to  the  hundredths  figure,  and  hence  find  the  true  square  root  to 
hundredths. 

Find  by  the  method  of  Problem  11  between  what  values  the 
square  roots  of  the  following  lie : 

12.    17.  13.    50.  14.    65.  15.   102.  16.    147. 

Eeduce  to  the  simplest  form : 

20.    Divide  1  +  V2  by  1  +  V2  -f  V3. 

Suggestion.  —  To  free  the  divisor  of  surds  it  will  be  necessary  to  multiply 
both  dividend  and  divisor  twice  when  the  divisor  is  a  trinomial  such  as  this. 
First  multiply  by  the  expression  obtained  by  changing  one  of  the  signs  in 
the  divisor.     Thus, 

1+V2       ^       (l  +  \/2)(l+\/2-\/3)       ^3  +  \^-V3-V6^^^^ 
1  +  V2+V5      (l+V2  +  >/3)(l  +  V2-V3)  2V2 

Divide : 

21.  V3+V5byl4-V3  +  V5.  25.    V5- V3  by  V5- V3- 1. 

22.  1+V2by  1-V2+V5.  26.    V^  by  V^  +  V^  +  V^. 

23.  2by  V2+V3  +  V7.  27.    Va- V&  by  Va+ V5- Vc, 

24.  V2by  V2  +  V3+V5.  28.    1  -  Vm  by  1  +  Vm4- V^. 


CHAPTER  XIV 

QUADRATIC   EQUATIONS 

144.  Quadratic  Equations.  —  As  shown  in  §  73  and  §  96,  some 
problems  may  be  expressed  and  solved  by  use  of  equations  of  the 
second  degree,  or  quadratic  equations  (see  §  57  for  definitions). 
The  equations  .discussed  in  §  73  and  §  96  were  of  very  special 
types,  and  the  methods  of  solution  there  shown  are  not  applicable 
to  all  quadratic  equations.  We  shall  now  discover  how  to  solve 
any  quadratic  equation  whatever. 

A  quadratic  equation  that  contains  a  term  of  the  first  degree  in 
the  unknown  number  is  called  a  complete  quadratic  equation.  One 
that  does  not  contain  a  term  of  the  first  degree  in  the  unknown 
number  is  called  a  pure  quadratic  equation. 

Thus,  3  w'-^  —  4  n  =  5  is  a  complete  quadratic  equation.  And  1 1^  —  9  =  0 
is  a  pure  quadratic  equation. 

By  simplifying  and  combining  similar  terms,  if  necessary,  any 
quadratic  equation  whose  unknown  number  is  denoted  by  x  can 
be  written  in  the  form  Ax^ -\-  Bx-\-  (7=  0,  where  A,  B,  and  C  are 
known  numbers. 

145.  Quadratics  solved  by  Factoring.  —  Some  quadratics  can  be 
solved  hy  factoring,  as  shown  in  §  96.  The  rule  there  given  should 
now  be  reviewed  and  applied  in  the  following  exercises. 

Note.  —  It  must  be  remembered  that  every  quadratic  equation  has  two 
roots.  Both  roots  should  be  checked  by  seeing  if  they  satisfy  the  given 
equation. 


270  ELEMENTARY  ALGEBRA 


EXERCISES 

Solve  by  factoring,  and  check : 

1.  a^==7a-V2. 

2.  P^  +  3P=10. 

3.  ^2  +  9^+14  =  0. 

4.  2a.-2  4-5a;  =  3. 

5.  6  F2+13  F+6=0. 

6.  10f-\-S^t  =  l. 


7.  6?/2  =  23?/  +  4. 

8.  2-R  =  21E^. 

9.  54^  =  9;s2_^72. 


10. 

4.A  =  A^-n, 

11. 

ml      1 

12. 

i_i_28      11 

13. 

15/r+4  =  -|. 

14. 

5  =  ^  +  f. 

r      7^ 

15. 

a2  +  i.a  =  3f. 

16. 

1  +  ^_6P=0. 

17. 

18. 

5r-8.a. 

19. 

¥s*— ■ 

20. 

lw--«- 

21 

3&_10     6+120 

14                h 

99 

1          aj-7       a;+24  _^ 

2a;-2     aj+1      5a^-5 

23. 

24. 

s+3_10         2    ^ 
2          3      s  +  3 

25. 

3(^-1)  _5  ,  2(^  +  1) 
^+1            '     ^-1 

26. 

r-1                    r+1 

27. 

3a  =  a2-18. 

146.  Quadratics  solved  by  Square  Root.  —  Any  pure  quadratic 
equation  can  be  solved  by  the  method  of  §  73. 

To  solve  a  pure  quadratic  equation  first,  by  clearing  of  fractions, 
transposing  terms,  etc.,  reduce  the  equation  to  the  form  x^  =  A,  where 
X  is  the  unknovm  number  and  A  is  known.  Take  the  square  root  oj 
both  members  of  the  resulting  equation,  attaching  the  double  sign  ± 
to  the  second  member. 

Example  1.  —  Solve  8  «2  =  2  «(« -  5)  +  10(«  +  2). 


QUADRATIC  EQUATIONS 


271 


Removing  parentheses,  transposing,  combining  terms,  and  dividing  by  the 
coefficient  of  «, 

Taking  square  root,  t=±  V^ 

These  roots  may  be  expressed  decimally,  if  desired.  Thus,  V30  =  5.477, 
approximately.    Hence,  «  =±  1.825,  approximately. 

Note.  —  In  some  problems  in  which  the  roots  involve  surds  it  is  sufficient 
to  express  the  roots  with  radical  signs.  But  in  many  applied  problems  it  is 
desirable  to  compute  the  approximate  values  of  such  roots  decimally. 

The  student  should  be  able  to  compute  the  roots  in  either  form  when 
desirable. 

Example  2.  —  Solve  Z>2  +  5  =  0. 

Transposing,  D^  =  -.  5. 

Taking  square  root,  D  =  ±  V—  5,  imaginary  numbern. 


EXERCISES 


Solve : 

1.  7y2-28  =  0. 

2.  10  F^- 150  =  4  V^. 

3.  7^  =  845-47^. 

4.  a(a  + 4)  =  4  a +  49. 

5.  (M+2){2M-{-3)=7  (M+S). 

6.  8««  +  72=:0. 

7.  5n2  +  20  =  0. 

8.  4.A^  =  7. 

9.  6i22_8  =  0. 

10.  5^2  +  10  =  0. 

11.  20  P2 4-8  =  0. 
X   _5 

125  ""a;* 

St^± 
4      3t 


12 


13.   ^  =  ^. 


14. 


15. 


16. 


17. 


18. 


19. 


20. 


21. 


y  +  1 
a2  +  4 


8 
F^  +  8 


,y-l 

6 
=  1. 


2 
9^2-1 


=  2. 


=  4. 


J 1,^9 

2Jc'     Tc"     4* 

6  5 

71  +  1 


=  L 


1 


w+1      n— 1 


=  3, 


272  ELEMENTARY  ALGEBRA 

22.    ^Aszl^Alll=:A±l,      26.    ^^+? 15-  =  0. 


23. 
24. 
25. 


^2-1       ^  +  1     ^-1  2  +  s       2s  +  3 

£±^ §_=i.  27.  2r+i+(^^+i)^^-^):^o 

6         Q  +  1  ^   ^  T4-2 

g  +  6      6^±l_n  28    2G^_l+^^3 
6^24-1     7+6""  '  "^     2(^  +  3     5 

4^4-1^8j9-19^  29.   ^l±lil±l-6  =  - 


3i)-4     7p-24  ^-1  a;  +  l 


Eind  approximate  to  hundredths  the  roots  of : 

30.  x^^l.  36.    5^^-14  =  0. 

31.  4a2-10  =  0.  37.   3«2=28. 

32.  8i2_i2  =  0.  ^^-    (P+4)(p-4)  =  4. 

33.  15n2-4  =  0.  39.    -1- =IZL§. 

r  +  3         4 

34.  7F^  =  26.  ^^^ ^^ 

35.  2i>2  =  i7.  4  iT-l        • 

147.  Quadratics  solved  by  completing  the  Square.  —  As  shown  in 
§  86,  any  trinomial  that  is  a  perfect  square  may  be  written  in 
either  the  form  a?  +  2  ah -\- h"^  ov  a?  —  2  ah -\-  h^.  Hence,  if  we  have 
given  either  the  binomial  a^  +  2  a6  or  a^  —  2  ah,  to  make  it  a  perfect 
square  we  must  add  6^,  i.e.  the  square  of  one  half  of  the  coefficient 
of  a  in  the  term  2  ah.     This  process  is  called  completing  the  square. 

Thus,  to  complete  the  square  whose  first  two  terms  are  n^  —  10  n,  we  add 
the  square  of  ^  of  10,  or  25.  This  gives  w^  —  10  w  +  25,  which  is  the  square 
of  w  —  5. 

By  use  of  this  process  of  completing  the  square  any  complete 
quadratic  equation  whatever  may  be  solved,  as  shown  in  the 
following  examples. 

Example  1.  —  Solve         v^  +  3  u  —  10  =  0. 

Transposing,  v^  -{-  S  v  =  10. 

Adding  the  square  of  ^  of  3  to  both  members,  to  complete  the  square  on 
the  left, 

«2  +  3  u  +  I  =  4,9 . 


QUADRATIC  EQUATIONS  273 

Taking  square  root,  »  +  f  =  ±  |. 

Transposing,  v  =  -  f  ±  |,  i.e.  -|  +  |or~|-|. 

Simplifying,  ?>  =  2  or  —  5 

Example  2.  —  Solve  6  P2  +  12  P  =  2. 

Dividing  by  5,  P2  +  i^  P  =  |. 

Adding  square  of  |  of  V,     P^  +  V  ^  +  if  =  If  • 

Taking  square  root,  P+  f  =  db  ^^46. 

Transposing,  P  =  —  |  ±  i  V56. 

If  the  approximate  values  of  these  roots,  expressed  decimally,  are  desired, 
they  may  be  found  by  taking  the  square  root  of  46  and  simplifying.  Thus, 
\/46  =6.782,  approximately.     Hence,  P=  0.15  or  -  2.55,  approximately. 

Example  3,  — Solve  y^  =  4y  —  7. 
Transposing,  y^  —  4y=  —  7. 

Adding  square  of  |  of  4,  ^/^  _  4  y  _}.  4  =  _  3, 

Taking  square  root,  y  —  2  =  ±  V—  3. 

Transposing,  y  =  2  ±  V— 3. 

These  roots  involve  imaginary  numbers. 

Evidently,  any  complete  quadratic  equation  may  be  solved  by 
the  following  rule  :  i^ 

(1)  Reduce  the  equation  to  the  form  x-  -\-px  =  q,  where  x  is  the 
unknown  number. 

(2)  Add  to  each  member  the  square  of  one  half  of  the  coefficient 
of  the  term  in  the  first  x>ower  of  the  unknovm  number. 

(3)  Take  the^square  root  of  both  members,  attaching  the  double 
sign  ±  to  the  second  member. 

(4)  Solve  the  resulting  linear  equations. 

EXERCISES 
Solve  by  completing  the  square : 

1.  2/2-62/  +  8  =  0.  6.  r^  +  a;  =  6. 

2.  n24-4w  =  12.  7.  F'-llF+30  =  0. 

3.  A^-^A  =  ^.  8.  d^  =  U-hd. 

4.  f  =  22  +  ^t.  9.  i1f2_^12JI^+15  =  0. 

5.  'ir'  +  14v  +  40  =  0.  10.  2A;2  +  A:  =  4. 


274  ELEMENTARY  ALGEBRA 


28.    -^+-^  =  1  + 


11.  2P2_5p  =  3, 

12.  6r2  +  6  =  13r.  '"  V^-^  '  i>  +  l      "  '  p-1 

13.  4TF^-llTr=3.  2^         1        I       2  0 

15.  3  2^2  =  17^4.28.  OA  Q         1  1 

30.  o —  = —  • 

16.  2b'-\-3b  =  4.,  y  +  2     y-2 

17.  ^lc'  =  k-\-l.  ^^         1       J       1       .       1 


18.  12  6^2^3  =  14  G^.  C^-1      C^-2      U-3 

19.  9^2^6^  +  26.  g^  c  +  2     4-c^o^ 

20.  16  a2_  96  a  =  1792.  '  c-1       2  c         ^* 

21.  iV^2_7j^_^i5^Q^  33^  (;2_5)2_,.(2;_10)2  =  37. 

22.  -^ ^  =  -1^.  34  n  +  3  ,r,-4_ 

9^        ^      I       ^     — ^  3  11 


4  +  r''^4-r'      3'  35.    _,_^+^^_^      ^ 

36. 


24.  ^-|+20^  =  42|.  _        1  1  1 


m  +  2     m  — 2      1  —  m 

25.  6ir  +  — :^- —  =  42.  2F-1  ,    F-l      3^-10 

±1  37. 1 = • 

26.  a^  +  7a;=7(a;  +  3)+4.  Y-^        Y-2         Y-3 

2^    ^  —  3      ^  +  4^2  3g       cg-4  (^-4    ^11 

'  w  +  5     w-7       '  '  2d-12     Sd-ie      2* 

Find  approximate  to  hundredths  the  roots  of : 

39.  w2  =  3n+7.  44.  12  Ii'  =  Ii  +  2. 

40.  u^-15u  =  60.  45.  34^  =  3«2_iq5^ 

41.  9^2^1-3A  46.  19  =  a+4a2. 

42.  />2  +  8Z)  +  ll  =  0.  47.  5M^  =  2M-{-7, 

43.  z^  +  6z  =  l.  48.  21-2v  =  4vl 

148.    Quadratics   solved   by  a   Formula.  —  Since   any  quadratic 
equation  may  be  written  in  the  general  form, 


QUADRATIC  EQUATIONS  275 

where  x  denotes  the  unknown  number  and  A,  B,  and   O  are 

known  numbers,  the  roots  of  this  equation  will  give  a  formula 

by  which  the  roots  of  any  particular  quadratic  equation  may  be 

written  down  at  once. 

Ax^  +  Bx  4-  (7  =  0    is    solved    by   completing    the    square    as 

follows : 

Dividing  by  ^,  a;^  +  —  x  +  -^  =  0. 

A        A 

B  C 

Transposing,  a;^  +  —  a;  =  — -  • 

A  A 

B  B^       B^  —  4:  AC 

Completing  square,    x^  + -r  ' 
A 

Taking  square  root, 
Transposing, 


2A 
Now,  by  replacing  A,  B,  and  C  in  the  formula  for  the  roots 


^"4^-2- 

4^2 

-^= 

^/B^-4AC 
2A 

x  = 

B       y/B^-4AC 
2A             2A 

^B±VB^-4AC 

_-B±^B--^AC 
^"  2A 

by  the  particular  values  which  they  have  in  any  given  equation, 
the  roots  of  any  quadratic  equation  may  be  written  down  at  once. 
Thus  the  long  process  of  completing  the  square  in  every  equation 
may  be  avoided.  Hence,  this  formula  should  he  mastered  and 
used  in  all  future  work  where  the  equation  cannot  be  readily  solved 
by  factoring. 

Example  1.  — Solve  3  n2  -  4  n  =  15.  , 

Written  in  the  form  Ax^  +  Bx  +  C  =  0,  this  becomes  *  ♦ 

3  n2  _  4  n  -  15  =  0. 

Here  ^  =  3,  J5  =  —  4,  and  C  =  — 15.     Substituting  these  values  for  A,  Is. 
and  C  in  the  formula,  

4  -j_  Vl(3  +  180 
n  = 

4  ±14 
=  3  or  -  |.  "*• 


276  ELEMENTARY  ALGEBRA 

Example  2.  —  Solve  4  F2  +  2  =  -  3  F. 
Transposing,  4  F2  +  3F-|-2  =  0. 

Here  ^  =  4,  5  =  3,  and  C  =  2. 


Substituting,  ^^-3  J:V9-32 

8 


_3_t-V-23    . 

s= ^- ,  imaginary  expressions 

8 


EXERCISES 

Solve  by  use  of  the  formula  • 

1.  /-32/  =  10.  ^^     2^n  +  l 

2.  i)2^10p  +  21=0.  *    ?^         2    * 

3.  2A^  +  A  =  Q.  22.    -^-^.Zd  =  0. 


4.   3^2_p5^  =  2. 


l-«         3 


5.    6/^2  +  ^-5=0.  23.    4  =  2+  ^ 


6.    7ii2  =  50?^_7. 


v^  2v 


7.  4Jf2_^3f+5  =  0.  24.    -  +  ^  =  11. 

8.  6ri2+6  =  13ri.  ^      .  4 
^                          25.   J?  +  3=      * 


9.  10a2  +  14  =  39a.  R-1 

10.  8i)2=65Z>-8.  2g    s_l  =  3_l. 

11.  2n^  +  Sn-ir4.  =  0.  '  s  3 

12.  5^2  =  22-7.  27.    2«  +  3^  a  +  3 

13.  15g2  +  3  =  g.  «-2       2a-l 

14.  ^  +  4  =  8JB.  28.    ^"•^  +  ^  +  ^  = 

^-4^^+3 

3a;2-4 

2a;  +  l' 

4F 


15.  2'?;2^7^_g 

16.  4fc^  +  5  =  6^.  29.   x  +  2 

17.  21c«=c  +  4. 

18.  9f=6  +  y.  30.   1+F=^^^ 

19.  12  +  a;  =  lla:2.  1_ 

20.  l-a2-3a  =  0.  ^^^   ^  +  6-^' 


QUADRATIC  EQUATIONS  277 

33.    -6- +1  =  3.  35.        ^  «  8 


^  +  1      A  x—4:     x  —  5     x—3 

36     ^-2^  +  2      10^-8_Q 


37. 


2  1  -  w      ?o  4- 1 


5w-10     w-\-2     w^^4: 


7-3M      1-M  ^23f+3. 

•  4-3Jf     2Jfcr+2     43f-2* 

•  18^2/4-1      22^  +  8 

8^4a;4-4     2a.'2-2  T^+1 

41.    8+Z  +  4-2P^8,  44^    (n-iy     n-2^g^ 

8-P     4  +  2P     3  2-4 

*2-   ^'-733-^'+-2~         ''^'    4+r      r-6-12 

Find  to  hundredths  the  roots  of  : 

46.  a2-5a  =  8.  48.    12(^2  =  3^4.19.    50.    6Q2  +  i  =  9Q. 

47.  4J\r2+^_7  =  0.    49.    9-2y  =  y\  51.    18  =  3 JS  +  2 -B^. 

149.  Problems  solved  by  Quadratic  Equations.  —  It  has  been 
seen  that  the  roots  of  quadratic  equations  may  be  fractions, 
negative  numbers,  surds,  or  even  imaginary  numbers^  In  solving 
a  problem  by  means  of  a  quadratic  equation,  it  is  necessary  to 
consider  the  nature  of  the  roots  to  see  if  both  roots  saU^y  all 
requirements  of  the  given  problem.  Thus,  a  problem  may  require 
for  its  solution  a  j9os?Y/ve  number.  If  one  of  the  roots  of  the 
equation  formed  is  found  to  he  negative,  it  must  be  discarded, 
although  it  satisfies  the  equation.  If  the  nature  of  a  problem 
requires  a  real  number  for  its  spluttqn,  and  the  roots  of  the  equa* 
tion  formed  are  found  to  \)Qi'*4((i(i^inary,  they  must  be  discarded. 


278  ELEMENTARY  ALGEBRA 

Example  1.  —  A  train  runs  at  a  uniform  rate  between  two  points  280  miles 
apart.  If  it  ran  5  miles  per  hour  faster,  it  would  make  the  distance  in  1  hour 
less  time.     Find  the  rate  of  the  train. 

Let  V  equal  the  rate  of  the  train. 

280 
Then  =  number  of  hours  required  for  the  run. 

V 

280 
And =  number  of  hours  required  at  the  increased  speed. 

„                                               280       280     ,  1 
Hence,  —  = +  1. 

V       V  +  5 

Solving,  t?  =  35  or  —  40. 

But  the  rate  must  be  a  positive  number.  Hence,  the  root  —  40  must  be 
discarded,  and  the  only  rate  possible  is  35  miles  per  hour. 

Example  2.  —  Find  the  real  number  whose  square  increased  by  82  equals 
8  times  the  number. 

Let  n  =  the  number. 

Then,  w^  +  32  =  8  w. 


Solving,  n  =  4  ±  V—  16. 

Since  both  roots  involve  imaginary  numbers,  the  problem  has  no  solution, 
i.e.  it  is  impossible. 


EXERCISES 

By  use  of  one  unknown  number  solve: 

1.  Eind  two  consecutive  arithmetical  numbers  tbe  sum  of 
whose  squares  is  313. 

2.  Eind  two  consecutive  arithmetical  numbers  whose  product 
is  240. 

3.  Divide  31  into  two  parts  the  sum  of  whose  squares  is  541. 

4.  rind  two  arithmetical  numbers  whose  sum  is  50  and  prod- 
uct 336. 

5.  Divide  14  into  two  parts  such  that  4  times  the  square  of 
the  larger  shall  exceed  6  times  the  square  of  the  smalkr  by  40. 

6.  The  denominator  of  a  certain  fraction  is  5  more  than  the 
numerator.  If  the  fraction  be  added  to  the  fraction  inverted,  the 
sum  will  be  2|-|. .   Find  the  fraction. 


QUADRATIC  EQUATIONS  279 

7.  A  pupil  was  to  divide  12  by  a  certain  number,  but  by  mis- 
fc^ke  he  subtracted  the  number  from  12.  His  result  was  5  too 
great.     Pind  the  number. 

8.  The  side  of  one  square  exceeds  that  of  another  by  3  inches, 
and  its  area  exceeds  twice  the  area  of  the  other  by  17  square 
inches.     Find  the  lengths  of  their  sides. 

9.  A  rectangular  field  is  96  ft.  longer  than  it  is  wide,  and  it 
contains  298,000  sq.  ft.     What  are  its  dimensions? 

10.  A  floor  can  be  paved  with  200  square  tiles  of  a  certain 
size.  If  each  tile  were  1  in.  shorter  each  way,  it  would  require 
288  tiles.     Find  the  size  of  each  tile. 

11.  In  the  center  of  a  rectangular  room  is  a  rug  9  ft.  by  12  ft. 

Around  this  is  a  border  of  uniform  width.     The  area  of  the  floor 
is  208  sq.  ft.     Find  the  width  of  the  border. 

12.  A  lawn  is  40  ft.  by  90  ft.    Two  boys  agree  to  mow  it.    The 

first  boy  is  to  mow  one  half  of  it  by  cutting  a  strip  of  uniform 
width  around  it.     How  wide  a  strip  must  he  cut  ? 

13.  A  farmer  has  a  field  of  wheat  80  rd.  wide  and  120  rd.  long. 
How  wide  a  strip  must  he  cut  around  the  field  in  order  to  have 
one  fourth  of  the  wheat  cut? 

14.  In  Problem  13,  how  wide  a  strip  must  he  cut  around  the 
field  in  order  to  have  one  half  of  the  wheat  cut? 

15.  A  farmer  has  a  field  40  rd.  wide  and  80  rd.  long  to  be 
plowed  and  planted  in  corn.  He  wishes  to  plow  4^  acres  the  first 
day  by  plowing  a  strip  of  uniform  width  around  it.  How  wide  a 
strip  must  he  plow  ? 

16.  The  hypotenuse  of  a  right  triangle  is  4  in.  longer  than 
one  leg  and  2  in.  longer  than  the  other.  Find  the  sides  of  the 
triangle. 

17.  One  side  of  a  rectangle  is  4  in.  longer  than  the  other,  and 
its  diagonal  is  20  in.     How  long  are  the  sides? 

18.  Two  men  start  at  the  same  time  from  the  intersection  of 
two  roads,  one  driving  south  at  the  rate  of  4  miles  an  hour  and 


280.  ELEMENTARY  ALGEBRA 

the  other-  west  at  the  rate   of  3  miles  an  hour.     In  how  many 
iiours  will  they  be  25  miles  apart? 

19.  Two  trains  are  100  miles  apart  on  perpendicular  roads,  and 
are  approaching  the  crossing.  One  train  runs  10  miles  an  hour 
faster  than  the  other.  At  what  rates  must  they  run  if  they  both 
reach  the  crossing  in  2  hours  ? 

20.  A  train  running  6  miles  an  hour  slower  than  usual,  due  tc 
a  storm,  was  1  hour  late  in  making  a  run  of  252  miles.  Find  its 
speed. 

21.  An  engineer  increased  the  speed  of  his  train  5  miles  an 
hour,  and  made  a  run  of  360  miles  in  1  hour  less  than  schedule 
time.     What  was  the  speed  when  running  according  to  schedule  ? 

22.  A  company  of  people  arranged  for  a  dinner,  for  which  they 
contracted  to  pay  %  45.  Five  of  the  people  were  unable  to  attend, 
and  as  a  result  each  of  the  others  had  to  pay  30  cents  more  than 
he  had  expected.     How  many  were  present? 

23.  A  farmer  sold  his  wheat  for  $540.  A  month  later  the 
price  of  wheat  advanced  10)^  a  bushel,  and  he  found  that  if  he 
had  held  his  wheat  until  that  time,  he  could  have  received  the 
same  money  and  have  had  75  bu.  left.  How  much  per  bushel  dic^ 
he  receive  for  the  wheat? 

24.  A  trader  bought  a  flock  of  sheep  for  $1020.  Two  of  them 
died,  and  he  sold  the  rest  at  a  profit  of  $3  a  head.  He  made 
$225  on  the  transaction.     How  many  did  he  buy? 

25.  A  man  bought  a  farm  for  $20,000.  Later  he  sold  all 
except  35  acres  of  it  at  a  gain  of  $  35  per  acre  over  the  cost,  and 
received  just  what  he  paid  for  the  whole  farm.  How  many  acres 
in  the  farm  ? 

26.  One  pump  can  fill  a  tank  in  6  minutes  less  time  than  ia 
required  for  another  pump  to  fill  it.  The  two  working  together 
can  fill  it  in  10^  minutes.  Find  the  time  required  for  each  pump 
alone  to  fill  the  tank. 


QUADRATIC  EQUATIONS  281 

27.  One  of  two  pumps  can  fill  a  tank  in  28  minutes,  and  the 
time  required  for  the  other  pump  to  do  it  is  19|-  minutes  longer 
than  is  required  for  the  two  pumps  together  to  fill  it.  Find  the 
time  required  for  the  two  pumps  together  to  fill  the  tank. 

28.  With  A's  help  B  could  do  a  piece  of  work  in  6J  hours  less 
time  than  would  be  required  for  B  to  do  it  alone.  A  could  do  it 
alone  in  21  hours.     In  how  many  hours  could  B  do  it  alone  ? 

29.  One  pipe  could  empty  a  tank  in  2  hours  less  time  than 
would  be  required  for  a  second  pipe  to  empty  it.  After  the  first 
pipe  has  been  open  4  hours,  it  is  closed  and  the  second  pipe  is 
opened.  The  second  pipe  finishes  emptying  the  tank  in  5  hours 
more.     In  what  time  cpuld  each  pipe  alone  empty  it  ? 

30.  An  aviator  flew  80  miles  against  the  wind,  and  back  again, 
in  6  hours.  His  speed  against  the  wind  was  10  miles  per  hour 
greater  than  the  speed  of  the  wind.  Find  the  speed  of  the  wind, 
also  what  his  speed  would  have  been  had  there  been  no  wind. 

31.  A  line  16  in.  long  is  to  be  divided  into  two  parts  such  that  the 
ratio  of  the  whole  line  to  the  longer  part  shall  be  equal  to  the  ratio 
of  the  longer  part  to  the  shorter  part.  Find  to  tenths  of  an  inch 
the  lengths  of  the  parts. 

Note.  —  The  line  in  Problem  31  Is  said  to  be  divided  in  "extreme  and 
mean  ratio."  Since  the  time  of  the  Greeks  the  problem  to  divide  a  line  in 
extreme  and  mean  ratio  has  been  called  the  *'  Problem  of  the  Golden  Section." 

32.  Experience  has  shown  that  a  book,  photograph,  front  of  a 
tall  building,  or  other  rectangular  object,  is  most  pleasing  to  the 
eye  when  its  length  and  width  are  obtained  by  dividing  their  sum 
in  extreme  and  mean  ratio. 

The  page  of  a  book  is  to  be  5  in.  wide.  How  long  should  it  be 
made  to  be  most  pleasing  to  the  eye  ? 

Suggestion.  —  If  w  denotes  the  length,  ^  "^     =  - . 

n         5 

33.  A  photograph  is  to  be  mounted  on  cardboard  6  in.  wide 
How  long  should  the  board  be  cut  ? 


282  ELEMENTARY  ALGEBRA 

34.  An  architect  is  designing  a  building  whose  frontage  on  the 
street  is  to  be  80  ft.  How  high  should  he  make  it  for  the  greatest 
beauty  ? 

35.  An  open  box  that  shall  hold  384  cu.  in.  is  to  be  made  from 
a  square  piece  of  cardboard  by  cutting  out  a  6-inch  square  from 
each  corner  and  turning  up  the  sides.  Find  the  size  of  the  piece 
of  cardboard  that  must  be  used. 

36.  An  open  box  that  shall  contain  324  cu.  in.  and  that  shall 
be  twice  as  long  as  wide  is  to  be  made  from  a  rectangular  piece  of 
tin  by  cutting  out  2-inch  squares  from  the  corners  and  bending  up 
the  sides.     Find  the  dimensions  of  the  piece  of  tin  required. 

37.  A  flat  disk  of  sheet  metal  3  inches  in  radius  is  to  be 
stamped  by  means  of  a  die  into  a  box  lid  1  inch  deep.  The  total 
area  of  the  lid  must  equal  the  area  of  the  flat  disk.  What  will 
be  the  radius  of  the  lid  ? 

Suggestion.  —  If  r  is  the  radius  of  the  lid,  its  total  surface  is  composed  of 
a  circle  whose  area  is  irr^  and  a  cylindrical  part  whose  area  is  2  irr.      Hence, 

38.  If  the  radius  of  the  disk  in  Problem  37  is  2  inches  and  the 
depth  of  the  lid  ^  inch,  what  will  be  the  radius  of  the  lid  ? 

39.  The  rim  of  the  front  wheel  of  a  carriage  is  2  ft.  less  than 
the  rim  of  the  hind  wheel.  The  front  wheel  makes  88  more 
revolutions  in  going  a  mile  than  the  hind  wheel.  Find  the  length 
of  the  rim  of  each  wheel. 

40.  The  circumference  of  the  little  wheel  on  a  sewing  machine 
is  29  inches  less  than  that  of  the  big  wheel  which  drives  it. 
While  the  belt  moves  a  distance  of  1275  inches,  the  little  wheel 
makes  116  more  revolutions  than  the  big  one.  Find  the  circum- 
ference of  each  wheel. 

41.  A  park  is  900  ft.  wide  and  1650  ft.  long.  It  is  desired  to 
move  out  the  boundary  lines  the  same  distance  along  one  side  and 
both  ends  so  that  the  area  inclosed  will  be  just  twice  as  great  as 
it  now  is.     Find  how  far  the  boundary  lines  must  be  moved. 


QUADRATIC  EQUATIONS 


283 


42.  The  longer  base  of  a  trapezoid  is  14  in.  longer  than  the 
other  base,  the  altitude  is  equal  to  the  shorter  base,  and  the  area 
is  120  sq.  in.  Find  the  lengths  of  the  bases 
and  the  altitude. 

43.  If  any  two  chords  AB  and  CD  of  a  circle 
meet  at  0,  then  AO  x  OB=COx  OD. 

If  00  =  3  in.,   OD 
find  AO  and  OB. 


2  in.,  and  AB  =  S  in., 


44.   In  Problem  43,  if  (70  =  9,  0D  =  7,  and,^B=22,  find  AO 
^_    and  OB. 


45.  AB,  the  span  of  the  stone 
arch  ACB  that  is  to  be  constructed, 
is  40  ft.  (^0  =  20  ft.  and  OB  =  20 
ft.),  and  the  diameter  CD  of  the 
arch  is  50  ft.  Find  CO,  the  rise  of 
the  arch. 

46.  If  the  span  of  an   arch    is 
16  ft.  and  the  diameter  20  ft.,  find  the  rise  of  the  arch 

47.   If  the  span  of  an  arch  is  20  ft.  and  the  diameter  30  ft., 
compute  the  rise. 


150.  Literal  Equations :  Formulae.  —  If  a  literal  equation  or 
formula  (see  §  113)  is  of  the  second  degree  with  respect  to  the 
unknown  number,  it  may  be  solved  by  the  methods  of  this 
chapter. 

Example.  —  Solve  t(t  —  b)  =a(a  +  6)  for  t. 

Removing  parentheses,      t^—  bt  =  a^  +  ab. 
Transposing,  t'^—bt-  (a^  +  ab)  =  0. 


In  formula,  A  =  l,  B  = 
Substituting  in  formula, 


6,     C=  -(,a^-\-ab). 


2 

_b±(2a-\-b) 
2 

=  a  +  6  or  —  a. 


284  ELEMENTARY  ALGEBRA 

EXERCISES 

1.  Solve  av?  =  h  for  n. 

2.  Solve  aV  —  a'*  =  0  for  v. 

3.  Solve  mn  =  m^  —  4  for  m. 

4.  Solve  2  o;^  —  3  r^  =  5  £cr  for  a; ;  for  r. 

5.  Solve  W'  +  2A'  =  3WA  for  TT;  for  ^. 

6.  Solve  y^  -{-2  my  —  n^-\-2  mn  for  y,  for  n. 

7.  Solve  v^  -{-2rw-\-  s  —  0  for  w. 

8.  Solve  z-\-  —  —  n-\-—  for  w :  for  z. 

n  z 

9.  SolveP2  +  2P  =  vP+2'yforP. 

10.  Solve  f-{-at-\-U-\-ah=^0  for  «. 

11.  Solve  A^—  (m  —  n)A  =  mn  for  ^. 

12.  Solve  :i±-^  +  i=^  =  1  forn;  for  B, 

1-Rn      1+Rn 

13.  Solve/S'  =  iw[2a4-(^-l)(^]  forn. 

14.  Solve /S^^^"^"-^^  forn. 

2 

15.  If  an  object  is  let  fall  downward  from  a  height  s  feet  above 
the  earth,  the  time  in  seconds  required  for  it  to  strike  the  earth 
is  computed  by  the  formula  s  =  16 1^.     Solve  for  t. 

16.  How  long  would  it  take  a  body  to  fall  a  distance  of 
1280  feet  ? 

17.  If  an  object  is  thrown  downward  toward  the  earth  with  an 
initial  velocity  of  v  feet  per  second,  the  distance  s  in  feet  that  it 
will  fall  in  t  seconds  is  computed  by  the  formula  s  =  vt-\-lQ>  f. 
Solve  for  t. 

18.  An  object  is  thrown  downward  with  a  velocity  of  32  feet  a 
second  from  a  distance  of  240  feet  above  the  earth.  How  long 
will  it  take  for  it  to  strike  the  ground  ? 


QUADRATIC  EQUATIONS  285 

19.   The  velocity  of  the  discharge  of  water  from  a  pipe  is  com- 
puted  by  the  formula  4 1;^  +  5  v  =  2  -{■— — .     Solve  this  for- 

mula  for  v, 

SUPPLEMENTARY  EXERCISES 

A  second  method  of  solving  a  quadratic  equation  by  completing 
the  square  besides  that  given  in  §  147  is  shown  below. 

1.    Solve  aa^  +  bx-\-c  =  0  for  x. 

Solution.  —  Multiply  through  by  ,4  a,  or  4  times  the  coefficient  of  x^. 
Then  4  a^x^  +  4  a6x  +  4  ac  =  0. 

Transposing,  4  a^x^  +  4  dbx  =  —  4  ac. 

Add  h^^  the  square  of  the  coefficient  of  x  in  the  original  equation,  to  both 
members. 

Then  4  aH^  +  4  aftx  +  6^  =  &2  _  4  ac. 


Taking  square  root,  2  ax  +  6  =  ±  y/h'^  —4  ac. 


Transposing,  2  ax  =  —  h±  y/h'^  -4ac 


Dividing  by  2  a,  x  =  -  &  =b -/^/^  -  4  ac^ 

2a 

Solve  by  the  method  of  Problem  1 : 

2.  Sx^  +  4:x  +  o  =  0,  7.   15F'-7  =  9F. 

3.  7w2_37i-4  =  0.  8.    662  =  6-1-5. 

5.  12  K'-{-K=  11.  10.   SM^==7M+15. 

6.  9a2  =  3a  +  7.  11.   6?-2  =  13r-6.' 


CHAPTER  XV 

SYSTEMS   INVOLVING    QUADRATIC    EQUATIONS 

151.  Systems  involving  Quadratic  Equations.  —  Some  problems 
may  be  expressed  and  solved  by  means  of  a  system  of  two  equa- 
tions containing  two  unknown  numbers  in  which  at  least  one 
of  the  equations  is  of  the  second  or  higher  degree.  In  this  chap- 
ter we  shall  discuss  only  systems  in  tchich  one  equation  is  linear 
and  the  other  quadratic,  these  being  suflB.cient  for  the  present  needs 
of  the  student.  A  complete  treatment  of  systems  involving  quad- 
ratic and  higher  equations  will  be  found  in  the  Second  Course. 

152.  Elimination  by  Substitution. — The  easiest  method  of 
eliminating  one  of  the  unknown  numbers  in  a  system  consisting 
of  one  linear  and  one  quadratic  equation  is  by  substitution.  See 
§128. 

Example—  Solve    f  »>*  -  2  n  =  5,  (1) 

1  m2  +  n2  =  10.  (2) 

Solving  (1)  for  m,  m  =  2n  +  6. 

Replacing  m  by  2  w  +  5  in  (2),  (2  ?i  +  5)2  +  ^^  =  10.  (3) 

Solving  (3),  w  =  -  1  or  -  3. 

Replacing  w  by  —  1  in  (1),  to  +  2  =  5, 

m  =  3' 
Replacing  w  by  —  3  in  (1),  m  +  6  =  5, 

m  =—  1. 
Hence  there  are  two  solutions  : 

VI  =  3,   n=—l,   or  m  =  — 1,    n=~-3. 

As  shown  in  the  above  example,  elimination  of  one  of  the  un- 
known numbers  in  a  system  containing  a  linear  and  a  quadratic 
equation  always  leads  to  a  quadratic  equation  in  one  unknown 
number,  which  may  be  solved  by  the  methods  of  Chapter  XIV. 


SYSTEMS  INVOLVING   QUADRATIC  EQUATIONS    287 

By  substituting  each  of  the  values  thus  found  for  one  of  the 
unknown  numbers  in  the  linear  equation  of  the  given  system,  a 
corresponding  value  of  the  other  unknown  number  is  found. 

Hence,  a  system  consisting  of  one  linear  and  one  quadratic  equa- 
tion lius  two  solutions. 


EXERCISES 


Solve : 


1. 

2  a  -f-  6  =  4, 

0"  +  ^  =  ^. 

13. 

\a^  +  f  =  7^, 
[3x-2y  =  l. 

2. 

.3x'-y'  =  23. 

14. 

2W-h2D  =  5  WJJy 

l2TF+2Z>=5. 

3. 

m  4-  ^  =  5, 
mn  =  4. 

15. 

2^^4-3^2  =  11  +  4^, 
U^5  =  3A;. 

4. 

IA-B  =  1, 
\AB=2, 

16. 

E'-EF-{-F^==  31, 

6. 

u^-\-uv-{-v^  =  S9, 
u  —  v  +  3  =  0. 

17. 

a4-6  =  4, 
6  +  a  =  ab. 

6. 

\V'  =  4.t, 
If+2^=4. 

18. 

'   W~3=Vy 

7. 

fr,2  =  4r2+18, 
l3ri  =  4r2  +  24. 

19. 

8F+12  =  4^, 
\3V'  +  2t'=^S-\-t. 

8. 

y  +  3i?5  +  g2  =  22, 
l2p  =  g. 

20. 

2x-5  =  x', 
.x-{-3x'  =  2xx'. 

9. 

2i?  +  r=:14, 
I  222  +  3  22,.  =  49. 

21. 

(6-2/)(7  +  ^)  =  80, 
■  z  +  y  =  5. 

10. 

|c  +  2(^  =  4, 

22. 

m=3w  +  l, 
,n^  +  mn  =  33. 

11. 

M-\-N=2, 
M'-MN-\-N'=6. 

23. 

\R  =  3S+1. 

12. 

y-2x  =  12, 
,2/2_ar2_3a;-}.6  =  0. 

24. 

3T-t-12  =  0, 

19^2  +  ^2^72. 

288  ELEMENTARY  ALGEBRA 


fa      &_13 
25.     U"^a       6'  28. 


a-{-h=n.  {G-\-t  =  L 


26. 


M+N=2y  [V.-l'^  —  'M 

-^4.1  =  6  ^^*     U^v"^' 


27.       -,^_g^l9,  30.  B     B'     B'' 

I        p~  p'  \a  +  B  =  9, 

31.  Find  two  numbers  whose  difference  is  2  and  the  sum  of 
whose  squares  is  34. 

32.  The  hypotenuse  of  a  right  triangle  is  20  feet  and  the  sum 
of  the  legs  28  feet.     Find  the  lengths  of  the  legs. 

33.  A  rectangular  field  is  40  rods  longer  than  it  is  wide,  and 
its  area  is  1152  square  rods.     Find  the  dimensions  of  the  field. 

34.  A  rectangular  field  contains  18  acres,  and  the  length  of  the 
fence  around  it  is  232  rods.  Find  the  length  and  width  of  the 
field. 

35.  The  base  of  a  triangle  is  5  inches  longer  than  the  altitude, 
and  the  area  is  42  square  inches.     Find  the  base  and  altitude. 

36.  Of  two  squares  the  side  of  one  is  7  feet  more  than  the  side 
of  the  other,  and  the  area  of  the  larger  is  161  square  feet  more 
than  the  area  of  the  other.     Find  the  sides  of  the  squares. 

37.  A  bin  is  to  be  made  to  hold  144  cubic  feet.  It  must  be  3  feet 
deep  and  3  times  as  long  as  wide.     Find  the  dimensions  of  the  bin. 

38.  Of  two  machines  in  a  mill  one  can  turn  out  an  order  of 
goods  in  3  hours  less  time  than  the  other,  and  if  both  machines  are 
operated  at  once,  they  can  turn  out  the  goods  in  2  hours.  How 
long  would  it  require  each  machine  alone  to  turn  out  the  goods  ? 

39.  If  one  machine  in  a  mill  can  produce  a  quantity  of  goods 
in  one  half  of  the  time  required  for  a  machine  of  smaller  capacity 
to  do  it,  and  the  two  machines  together  can  produce  the  goods 
in  4  days,  how  many  days  would  it  take  each  machine  alone  to 
produce  the  goods  ? 


SYSTEMS  INVOLVING   QUADRATIC  EQUATIONS    289 

40.  If  one  boiler  would  consume  a  tank  of  water  in  2  hours 
less  time  than  another,  and  the  two  together  would  exhaust  it 
in  2  hours  55  minutes,  how  long  would  the  tank  of  water  supply 
each  boiler  alone  ? 

41.  In  any  circle  whose  radius  is  R,  diameter  Dj  and  area  Sj 

S  =  7VR\ 

D  =  2R. 

By  eliminating  R  between  the  two  equations,  get  the  equation 
between  S  and  D. 

42.  In  any  sphere  whose  radius  is  Rj  diameter  D,  and  area  S, 

D  =  2R. 

By  eliminating  R  between  these  two  equations,  derive  the 
equation  between  /S  and  D. 

43.  In  any  right  circular  cylinder  the  radius  of  whose  base  is 
R,  altitude  H,  area  of  cylindrical  surface  S,  and  volume  V, 

S  =  7rRII, 
V  =7r  R'H. 

Eliminate  R  between  these  two  equations  and  get  a  formula 
between  S,  V,  and  //. 

153.  Graphs  of  Quadratic  Equations.  —  It  was  shown  in  §  124 
that  the  graph  of  a  linear  equation  containing  two  unknown 
numbers  is  always  a  straight  line.  Similarly,  it  will  be  found 
that  in  general  the  graph  of  a  quadratic  equation  containing  two 
unknown  numbers  is  some  kind  of  curve. 

For  example,  consider  the  graph  oi  y'^  =  Q  x  -\-  IQ. 

By  assigning  values  to  x,  and  computing  the  corresponding  approximate 
values  of  y,  the  following  sets  of  values  of  x  and  ij  are  obtained  : 


X 

30 

20 

10 

6 

2 

1 

0 

-1 

-n 

Greater  neg. 
values 

y 

±16.9 

±14 

±  10.3 

±8.4 

±5.8 

±5 

±4 

±2.6 

0 

Imaginary 

290 


ELEMENTARY  ALGEBRA 


By  locating  the  points  corresponding  to  these  sets  of  values  of  x  and  y  and 
joining  them,  as  in  §  124,  we  get  the  curve  shown  below. 


^'nse^ve  that  the  graph  is  perfectly  symmetrical  with  reference  to  the  axis 
X-3^.  By  assigning  greater  and  greater  positive  values  to  x,  corresponding 
seal  vames  can  always  be  found  for  y,  which  shows  that  the  two  branches  of 
the  curve  continue  indefinitely  to  the  right.  But  if  negative  values  are 
assigned  xio  x  larger  than  1|,  the  corresponding  values  of  y  are  found  to  be 
imaginary,  which  shows  that  no  part  of  the  graph  extends  farther  to  the  left 
than  x  —■-   1|. 

Thfc)  ;ibove  curve  is  called  a  parabola.  It  is  the  kind  of  path  in 
which  astronomers  have  found  that  many  comets  move.  Other 
comets  and  all  of  the  planets  move  in  closed  curves  called  ellipses. 

154.  Systems  solved  Graphically.  —  The  solution  of  a  system 
consisting  of  one  linear  and  one  quadratic  equation  may  be  found 
graphically,  as  in  §  132,  by  drawing  the  graphs  of  both  equations 


SYSTEMS  INVOLVING   QUADRATIC  EQUATIONS    291 


upon  the  same  axes.  The  sets  of  values  of  the  unknown  numbers 
corresponding  to  the  points  where  the  graphs  meet  are  the  solu- 
tions of  the  system. 

-y  =  4,  (1) 

41/2  =  400.  (2) 

The  graph  of  (1)  is  the  straight  line  (1)  in  the  figure  below.     By  assign- 
ing values  to  x  in  (2)  the  following  sets  of  values  of  x  and  y  are  obtained  : 


Example.  —  Solve  graphically 


X 

0 

5 

-5 

10 

-10 

15 

-15 

20 

-20 

y 

±10 

±9.7 

±9.7 

±8.7 

±8.7 

±6.6 

±6.6 

0 

0 

For  either  positive  or  negative  values  of  x  greater  than  20,  y  is  imaginary. 
And  for  either  positive  or  negative  values  of  y  greater  than  10,  x  is  imaginary. 
Hence  the  graph  lies  entirely  within  the  region  bounded  by  a;  =  20  on  the 
right,  X  =  —  20  on  the  left,  y  =  10  above,  and  y  =—  10  below.  The  graph  of 
(2)  is  seen  to  be  the  ellipse  (2)  in  the  figure. 


292 


ELEMENTARY  ALGEBRA 


The  straight  line  (1)  meets  the  curve  (2)  at  two  points,  P  and  Q.  The 
set  of  values  x  =  12,  ?/  =  8  corresponding  to  P,  and  the  set  of  values  x—  —  5.6, 
y  =  —  9.6  corresponding  to  Q,  are  the  two  solutions  of  the  system,  and  will 
be  found  to  satisfy  both  equations. 


EXERCISES 


Draw  the  graphs  of : 

1.  0^2  =  4?/ +  4. 

2.  0^  +  2/2  =  256. 

3.  40^  +  92/2  =  324. 

Solve  graphically: 

ix-y  =  2, 


4.  x'-y^  =  16, 

5.  9T'-y^  =  S6. 

6.  xy-^Sx-10y=:S0, 


7. 


9. 


a;2  +  2/2  =  100. 
y-2x  =  2, 
4a;2  +  2/2  =  164. 

a^  =  22/  +  l, 
a;  +  2  2/  =  4. 


10. 


11. 


12. 


5  0^  +  22/2  =  532, 
3o;-2/  +  10  =  0. 
2x-5y  =  S, 
o.-2_2/2  =  171. 
2  a;  -  3  2/  =  10, 
3i»2_5a;2/  =  2  2/2. 


13.  If  the  two  graphs  of  a  system  did  not  meet  at  all,  what 
kind  of  solutions  would  the  system  have  ? 

14.  If  the  graph  of  the  linear  equation  of  a  system  just  touched 
the  graph  of  the  quadratic,  what  would  be  the  nature  of  the 
solutions  ? 


SUPPLEMENTARY  EXERCISES 

The  roots  of  a  quadratic  equation  in  one  unknown  number  may 
be  obtained  graphically  as  follows : 

r       O 

X^  ~~  ?/ 

Eliminating  y  from  the  system  J  '  _^  gives  the  gen- 

eral quadratic  equation  in  one  unknown  number  ax^  +  6ic  +  c  =  0. 
Since  the  roots  of  this  equation  must  satisfy  both  equations  of 
the  above  system,  they  may  be  found  by  solving  this  system 
graphically. 


SYSTEMS   INVOLVING   QUADRATIC  EQUATIONS    293 


1. 

-20 


Solve 
=  0. 


2a;2-f3a; 


Solution.  —  This  equa- 
tion corresponds  to  the 
system 


2y  +  3a;-20  =  0. 


(1) 

(2) 


The  graph  of  (1)  is  the 
parabola  in  the  figure.  The 
graph  of  (2)  is  the  straight 
line  meeting  the  parabola  at 
P  and  Q.  The  values  of  x 
corresponding  to  the  points 
P  and  Q  are  2^  and  —  4, 
respectively,  which  are  the 
roots  of  the  given  equation. 

Note.  —  Since  the  equa- 
tion x^  =  y  will  be  the  same 
for  the  systems  formed 
from  all  quadratics,  the 
parabola  need  be  dra^vn 
only  once  and  all  quadratics 
solved  by  use  of  the  one 
figure.  To  solve  any  quad- 
ratic in  x,  substitute  y  for 
0:2,  and  find  two  points  of 
the  graph  of  the  linear 
equation  formed.  Connect- 
ing these  by  a  ruler,  ob- 
serve  the  values  of  x  corre- 
sponding to  the  points 
where  the  ruler  crosses  the  parabola.     These  are  the  roots  of  the  quadratic. 

Draw  a  large  perfect  parabola,  and  by  use  of  it  and  a  ruler  find 
the  roots  of: 


2.  x^  —  x--2  =  0, 

3.  a^~2a;=8. 

4.  sc'^^x  +  lO. 

6.  Q^^7x  +  X2==0. 


6.  a^  +  a;  =  12. 

7.  a;2  =  2a^  +  15. 

8.  2a^-5a:-|-2=0. 

9.  2a?2-}-5a;  =  3. 


10.  5x^=1  4- 4  a;. 

11.  6a;2  =  aj  +  35. 

12.  4.x-2=21a;-f  18. 

13.  5  ar^  =  3  a; -I- 14. 


GHAPTEK  XVI 

EXPONENTS 

155.  Positive  Integral  Exponents.  —  A  positive  whole  number 
used  to  indicate  how  many  times  a  given  number  is  to  be  used  as 
a  factor  was  defined  in  §  5  as  an  exponent.  Expressions  some- 
times are  encountered  in  which  negative  numbers,  zero,  and  frac- 
tions are  written  in  the  form  of  exponents.  In  this  chapter  we 
shall  discover  the  meanings  of  such  expressions. 

In  the  following  sections  are  given  some  laws  which  hold  for 
positive  integral  exponents.  Some  of  these  laws  have  been  used 
in  earlier  chapters,  and  are  given  here  primarily  for  review. 
Others  have  been  applied  only  in  special  cases. 

156.  Law  of  Exponents  in  Multiplication.  —  The  law  of  exponents 
in  multiplication,  as  stated  in  §  48,  is  expressed  in  symbols  by 

fl'"  X  a"  =  a'"'*"". 

The  general  proof  is  as  follows : 

a"*  =  axaxa"- to  m  factors. 
a"  =  axaxa--- to  71  factors. 

Hence, 

a"*  X  a"  =:(a  X  a  x  a  ...  to  m  factors)  (a  X  a  X  a  •••  to  n  factors) 
=  axaxaxa"' to  m-\-n  factors 
=  a"*"*"",  by  definition  of  an  exponent. 

EXERCISES 

Give  orally  the  products  of : 

1.  a^  X  a*.  3.   a^  X  a^.  B.    v  X  v^. 

2.  N^xN".  4.    «2o  X  ^^  g    ^12  ^  ^w 

294 


EXPONENTS  295 

7.  r^**  X  r".  12.   F^  X  F'  X  F^.        17.    a"+i  x  a""^  x  a^^. 

8.  y^^xy^.  13.   s^"xs'xs2.  18.   p^-^y^p^+\ 

9.  P^  X  P^'.  14.   QxQ^x  Q\         19.   6i2-«  X  6»+3. 

10.  d«  X  d^''.  15.   A;i2  X  A;!^  X  A;*'.        20.    F"""*  X  F^^+^m^ 

11.  a?xa^xa\        16.    a;"  x  a^"  X  a:^".         21.    Q^^"  X  Q*"-*. 

157.   Law  of  Exponents  in  Division.  —  The  law  of  exponents  in 
division  was  given  iu  §  52.     In  symbols, 

Qtri  ^  qh  -~  Qin-n^ 

This  follows  from  §  156,  because  the  quotient  a"*"**  times  the 
divisor  a**  equals  the  dividend  a"*. 

EXERCISES 


Give  orally  the 

quotients 

of: 

1.  /--y. 

8. 

(r^(^'- 

15. 

a.4n+l  _^  ^ 

2.    A'^^A'. 

9. 

^22  ^^W 

16. 

^2x+4^^x+l 

3.    iv^-h  w. 

10. 

mi^H-7ft". 

17. 

yAt+10   _^   yS 

4.    B^'^R'. 

11. 

B^-^B'^ 

18. 

pr3a+26  _j_   y^a+b^ 

5.    N'*^]Sr'. 

12. 

Jc'^^l''. 

19. 

Zm  _^  ^m-2^ 

6.      ^20^^12^ 

13. 

W"^  --  W\ 

20. 

Qt+^^Qt-2^ 

7.    x^'^x^. 

14. 

R'^^-^R^. 

21. 

Kr^^^K^-\ 

158.   Power  of  a  Power.  —  The  law  by  which  a  power  of  some 
number  is  itself  raised  to  a  power  is  expressed  by. 

That  is,  the  mth  power  of  the  nth  power  of  any  number  equals  tht 
mnth  power  of  the  number. 
For,  by  definition  of  an  exponent, 

(a")*"  =  a"  X  a"  X  a"  •  •  •  to  m  factor* 

_  ^n+n+n+  .••  to  m  terms^  j^y   §    -j^gg 

Thus,  (a^y  =  a2o ;  (i^)^  =  wjis .  (^4)9  ^^m 


296  ELEMENTARY  ALGEBRA 


EXERCISES 


Give  orally : 

1.   {aj. 

6. 

(f)\ 

11. 

(Ay. 

16. 

(QyK 

2.    {p^)\ 

7. 

(B-y. 

12. 

(s^y. 

17. 

(r^y-'. 

3.    {Py. 

8. 

Q^y, 

13. 

(ay. 

18. 

(R-+y- 

4.    {ry. 

9. 

(E!^y\ 

14. 

(w»)2- 

19. 

(^Jj2,n-ny 

5.  {y^y. 

10. 

(mY- 

15. 

(a^y-. 

20. 

(2y. 

159.  Power  of  a  Product.  —  The  law  by  which  a  power  of  the 
product  of  two  numbers  is  found  is  expressed  by, 

(aby  =  a"b". 

That  is,  tJie  nth  power  of  the  product  of  two  numbers  equals  the 

jproduct  of  the  nth  powers  of  the  numbers. 

For,  by  the  meaning  of  an  exponent, 

(aby  —  (ab)(ab)(ab)  •••  to  n  factors 

=  (ct  X  a  X  a  •••  to  n  factors)(6  X  6  X  ?>  •••  to  n  factors) 
=  a'^6^ 

By  similar  reasoning  the  law  can  be  shown  to   hold  for  any 
number  of  factors. 

Thus,  {ahcdy  =  a^h^cH^ ;  (8  xyY  =  3*  x^y^  =  81  oc^yK 

160.  Power  of  a  Fraction.  —  The  law  by  which  a  fraction  is 
raised  to  a  power  is  expressed  by, 

fa\" ^  a^ 
\bj       b"' 
That  is,  the  nth  power  of  a  fraction  equals  the  nth  power  of  the 
numerator  divided  by  the  nth  power  of  the  denominator. 


For,  /^lY  =  ^x-X-.-.ton  factors 

\bj       b      b      b 


Thus.  /2a6y^  (206)4 

'  X^cdJ       (3  cdy 


ax  a  X  a  '•'  to  n  factors 

b  xb  X  b  -"to  71  factors 

or 

b^' 

(2  ahy  _  2*  g^M  ^  16  a^b^ 
3*c4d*     SI  c^d* 


EXPONENTS 


297 


161.  Power  of  any  Monomial.  —  By  use  of  the  laws  of  exponents 
in  the  preceding  sections,  any  power  of  any  monomial  may  be 
obtained. 


Thus, 


And 


(2  a3&5c4)6  =  26(a8)6(65)6(c4)6  by  §  159 
=  G4  ai8630c24  by  §  158. 


81m^ 
625^12^20 


by  §§  159  and  168. 


EXERCISES 


Give  orally: 

1.  {2ay. 

2.  (p^f)'. 

3.  (2N^]\PY' 

4.  {a^hhy^ 

5.  {P'fiff. 


6.  (m'vhy. 

7.  {BA^B'y. 
3.  {m'nYy\ 

9.  (3  v'py, 

10.  (J^nY- 


11. 


12. 


©■■ 


2E^. 


"■  (^0' 


15 


fN'M'\ 


; 


162.   Root  of  a  Power.  —  To  find  any  root  of  any  power  of  a  hose 
divide  the  exponent  by  the  index.     That  is, 


Va^ 


m 

an  ' 


For,  by  §  158,  (a^)"  =  a**,     or  a*. 

Thus,  \^  =  a^;  \/^  =  c«  ;  Vm^  =  m^. 


EXERCISES 


Give  orally : 

1.    </^. 

6. 

</P^, 

11. 

^^. 

16. 

V^. 

2.    ^y". 

7. 

V¥, 

12. 

^F. 

17. 

Vw^. 

3.    -s/N'\ 

8. 

Vw^. 

13. 

■v//^^. 

18. 

VW. 

4.  ^e/^. 

9. 

^/?■^. 

14. 

^^. 

19. 

'Vg^, 

6.    ^. 

10. 

V^. 

15. 

■^^. 

20. 

^mi«». 

298  ELEMENTARY  ALGEBRA 

163.  Negative  Exponents.  —  If  the  law  of  exponents  in  division 
be  assumed  to  hold  in  case  the  exponent  of  the  divisor  is  greater 
than  that  of  the  dividend,  its  application  leads  to  a  new  kind  of 
number  symbol  called  a  negative  exponent. 

Thus,  a^^a^  =  a^-i  =  ^-3  j  ^e  ^  ^ii  =  ^-n  =  ^-5. 

The  meaning  of  a  negative  exponent  is  shown  as  follows : 
By  §  159,  a^^a^  =  a'-'  =  a-'. 

But  by  reduction,      a^  -f-  a^  =  —  =  — . 

Hence,  a"^  =  —  • 

In  general,  a~"  =  —  • 

a" 

That  is,  a  negative  exponent  indicates  a  power  of  a  base  that  is  to 
he  used  as  a  divisor. 

It  is  assumed  here,  and  can  be  proved,  that  all  of  the  laws 
that  hold  for  positive  integral  exponents  apply  also  to  negative 
exponents. 

164.  Zero  Exponent.  —  If  any  power  of  a  base  is  divided  by  it- 
self by  the  law  in  §  159,  it  leads  to  a  zero  exponent. 

Thus,  ««  -J-  a«  =  a6-6  =  a^  .^  t^  ^t^  =  «8-8  =  ^. 
In  general,  a"  -?-«**=  a"~"  =  a^. 

But  a*'H-a"  =  l. 

Hence,  a°=l. 

That  is,  any  base  with  the  exponent  0  equals  1. 

165.  Fractional  Exponents.  —  If  the  law  in  §  162  be  assumed  to 
hold  in  case  the  index  of  the  root  is  not  an  exact  divisor  of  the 
exponent,  its  application  leads  to  a  new  kind  of  number  symbol 
called  a  fractional  exponent. 

Thus,  \/^  =  a^;  v^  =  A 

n 

In  general,  a*"  =  Va". 


EXPONENTS  299 

That  is,  a  fractional  exponent  indicates  a  root  of  a  power  of  a 
number,  the  numerator  indicating  the  power,  and  the  denominator 
indicating  the  root. 

it  may  be  proved  that  the  laws  that  apply  to  positive  integral 
exponents  apply  also  to  fractional  exponents. 

EXERCISES 

Simplify : 

1.    n-2  X  n-\  14  ph  X  pf^  26.  (m"*)-®. 

15.  t^XtK  27.  (a^)K 

16.  K^xK^,  28.  (D^)K 
'    17.  nUnK  29.  (k^)K 

18.  V^^Vl  30.  (i^T^)! 

19.  bUeK  31.  (FV- 

20.  b^UbK  32.  V^. 

21.  L^^lK  ^3-  ^ 

22.  (O-^  3*-  "^^"'*- 

23.  (aO"'.  35.  ^l 

24.  (TT-O*.  ,  _ 

25.  (0-^  ^^'        ^  • 
SUPPLEMENTARY  EXERCISES 

Find  the  value  of : 

1.   8i             2.   8li  3.    64i             4.   2"^             5.   3-^. 
Simplify : 

6.  n^Xn-K                 11.  (a-2  +  3a-^-4)(a-i-2). 

7.  A^-^A-K               12.  (aj-i-l)(a;-2  +  a;-i  +  l). 

8.  (k-^'^yK                13.  (r^  +  l)-(ri  +  l). 

14.    (m-2  4-2m-^7i-*  +  0^(wi"*  +  ?^"^) 


2. 

^-5  ^  ^-8^ 

3. 

W^^  X  W~^. 

4. 

p-'xjf. 

5. 

r-^  X  r-\ 

6. 

N'-i-N-\ 

7. 

y-'^y-\ 

8. 

D-'^D^. 

9. 

-^--12  _;_  pr_j 

10. 

k'^k-\ 

11. 

r-5-  !r-\ 

12. 

a^  X  ai 

13. 

a;^  X  ici 

<^l 


10.  16-i.  15.  {N-'-iy. 


300  ELEMENTARY  ALGEBRA 

MISCELLANEOUS   EXERCISES 

1.  Find  the  value  of  W  in  the  formula  W=  ^-^j  when  w  = 

1800,  h  =  26,  and  k  =  0.75. 

2.  Find  the  value  of  A  in  the  formula 


A  =  Vs(s  —  a)(s  —  b){s  —  c), 
when  s  =  140,  a  =  80,  &  =  90,  and  c  =  110. 


3.  Simplify  the  expression  i  =  V 2  ?^  —  r ^4  r^  —  (rV2)2  and 
thus  find  a  simple  formula.     Then  find  L  when  r  =  12. 

4.  Solve  for  x  in  the  eq nation =  ^  ~^  .     Then  find  the 

ic  +  d         r 

value  of  X  when  d=24,  r  =  14,  and  ?•'  =  8. 

5.  Solve  F=^^^  for  each  letter  involved. 

6.  Solve  E  = for  n,  L,  and  ty. 

E  E 

7.  In   the  formulae   C=—  and  C"  = ,  eliminate  R  and 

express  E  in  terms  of  the  remaining  letters. 

8.  Given  P=        ^^^       .find   the   value   of  P  when   n  =  20, 

^  =  28,  and  m  =  16, 

9.  In  a^  ==  6^  +  c^  —  2  6m,  solve  for  m. 

10.  In  2  a;  =  ^ ^—  a;,  solve  for  x. 

11.  The  volume  of  a  hollow  sphere  is  found  by  the  formula 
V=^Tr{7^  —  r^^)y  where  r  is  the  radius  of  the  sphere  and  r'  the 
radius  of  the  hollow.  Find  the  weight  of  a  hollow  sphere  of 
brass  8  inches  in  diameter  when  the  diameter  of  the  hollow  is  7 
inches.     (A  cubic  inch  of  brass  weighs  0.303  lb.) 


MISCELLANEOUS  EXERCISES  301 

12.  The  area  of  a  ring  is  found  by  the  formula  ^  =  7r(r^  —  r'^, 
where  r  is  the  radius  of  the  outer  circle  that  bounds  the  ring  and 
r'  is  the  radius  of  the  inner  circle.  When  the  area  of  a  ring 
whose  outer  diameter  is  30  inches  is  91.1064  square  inches,  find 
the  inner  diameter. 

Factor  the  following  expressions : 

13    16  a^  -  12  a6  + 4  ac.  '20.    lQii^y-2y^. 

14.   4a2+i2a6  +  962.  21.    6  m^  + 12  m^a;  +  6  ma^. 

^         ^  22.   ad  -{- ay -^  dx -\-  xy. 

16.  25m*n  +  10mV  +  7i«. 

17.  64r'i2_i6^^2^ 

18.  16/-1.  24.   r^-i^r  +  J. 

19.  m*w-?i*.  25.    2/^  +  3Jy«  +  f  «*. 

26.   Reduce  ^^-^^^  — ^  to  its  lowest  terms. 
Sa^-24a;-9 

9.T.   Eeduce  ^^"*~^^'"^    to  its  lowest  terms. 
a^hx  —  Iroif 

A  rj,a 2  3* 

28.  Reduce  — — — =^^—  to  a  mixed  expression. 

2af-x-\-l 

29.  Reduce  ^'{^* -  n*) +S(m' -  mn^  ^ 

m^{m^  —  n^) 

30.  Change  1  +  ^  +^  ~  ^^  to  the  form  of  a  fraction. 

2Z>c 

31.  Change  x^  —  ^x-  ^  ^^^  ~  ^)  to  a  fractional  form. 

a;— 2 

32.  Change  ah,  ^~~    ,  and  ^"^     to  fractions  having  a  common 

a  +  6  a  — 6 

denominator. 

33.  Add2a,  3a  +  ^,   and  a-  —  - 

5  9 

34.  Add  ^-* ,  __A±^ ,  and  ^+» 

(6-e)(c-a)    (c-a)(a-6)  (a-6)(6-c) 


802  ELEMENTARY  ALGEBRA 

5  1  24 


35.    Simplify 


2(0;  + 1)      10(a:-l)      5(2  a; +  3) 


ax 


36.  Multiply  -^  by 

4:  ax       my""-^ 

37.  Multiply  a;^  -  a;  + 1  by  ^  +  -  + 1. 

or     X 

1  —  x^     1  —  v^ 

38.  Find  the  product  of ,   ^,  and  1-}- 

1  +y      x-\-xr 

39.  Divide  x* by  oj^  +  —  • 

x^  or 

40.  Simphfyr__-,-^3J^^-±^  +  -^^ 

1 

41.  Simplify      


X 


j,^        a     ^  trr  5  X  —  4:         S  X -{- 4:  OA  307-4-8  - 

42.  Solve  17  a; -^ —  =  20  a; ^ 5. 

A^     cs  1       a  —  x     4a  — X  ■.  o 

43.  Solve  • =  a  —  6  for  a;. 

h  c 

44.  Solve  — = 1 ^-  = for  x, 

ah  —  ax     he  —  ox     ac  —  ax 


45      |3a;+72/  =  33,  ^g     |6a  +  56  =  112, 

t2a; 


Solve  the  following  systems : 

+  72/  =  33, 

+  42/  =  20.  '"  •[8a-26  =  80. 

46      f7a;  +  32/  =  62,  43      f  6  ^  +  11  ^^  =  115, 

l5a;-22/  =  36.  '     j  8  ^-22?^  =  -30. 

^^      .12a;  +  82/  =  116,  50      f    2^+    3^  =  47 


ri2a; 
1    2a;- 


2/  =  3.  tl0;2-12^=-63. 

a;     y 

51.    Solve  ^l  for  x  and  «. 

a;  ,  y  ^ 

c      d 


MISCELLANEOUS  EXERCISES  303 

Solve : 

r   8a:-92/-7z=-36,  nx-\-A.y-    z  =  78, 

52.       12a?—    2/-32!=36,  53.    i  4a;-5  ?/ -3^  =  - 21, 

[   Q>x-2y-    2  =  10.  i    a;-32/-42  =  -37. 

54.  When  the  cost  of  an  article  to  a  merchant  is  lowered  20  %, 
the  merchant  who  keeps  the  same  selling  price  makes  30  % 
more.     Find  his  former  per  cent  of  gain. 

55.  By  getting  a  discount  of  5  %  off  for  cash  in  buying,  a  mer- 
chant makes  a  net  profit  of  8  %  more.  Find  his  rate  of  gain 
when  no  discount  is  received. 

56.  A  merchant  has  two  kinds  of  grain,  one  at  60  cents  per 
bushel,  and  the  other  at  90  cents  per  bushel,  of  which  he  wishes 
to  make  a  mixture  of  40  bushels  that  may  be  worth  80  cents  per 
bushel.     How  many  bushels  of  each  must  he  use  ? 

57.  A  merchant  has  three  kinds  of  sugar.  He  can  sell  3  lb. 
of  the  first  quality,  4  lb.  of  the  second  quality,  and  2  lb.  of  the 
third  quality,  for  60  cents;  or,  he  can  sell  4  lb.  of  the  first 
quality,  1  lb.  of  the  second  quality,  and  5  lb.  of  the  third  quality, 
for  69  cents ;  or,  he  can  sell  1  lb.  of  the  first  quality,  10  lb.  of  the 
second  quality,  and  3  lb.  of  the  third  quality,  for  90  cents.  Find 
the  price  of  each  quality. 

58.  How  much  cream  testing  36  %  butter  fat  must  be  mixed 
with  20  gallons  of  milk  testing  3  %  butter  fat  to  have  milk  test- 
ing  4.8  %  butter  fat  ? 

59.  Find  the  square  root  of  4  a*  — 16  a''  +  24  a^  — 16  a  +  4 

60.  Find  the  square  root  of  m^  +  2  m  —  1  —  —  +  — . 

61.  Find  the  square  root  of  33.1776. 

62.  Find  to  three  decimal  places  the  square  root  of  2. 

Simplify  the  following: 


63.    V96  a^a^.  64.    -s/ {x"  -  f){x  +  y). 


304  ELEMENTARY  ALGEBRA 

65.    xP^^.  66.    ^-±l^E^. 

67.  Multiply  4  a^y/H  by  f^A. 

68.  Multiply  a  +  2 v&  by  a  —  2 V&. 

69.  Divide  V20  +  Vl2  by  V5  +  V3. 

70.  Solve  4  ar^  + 16  a;  =  33. 

71.  Solve  4  a; -5^^  =14. 

a;  +  l 

72.  Find  to  three  decimal  places  the  roots  of 

^2^1     3 
2     X     ^      X 

73.  A  lady  finds  that  there  are  just  85  square  yards  of  floor 
surface  to  be  covered  in  two  of  her  square  bedrooms,  one  of 
which  is  3  feet  longer  than  the  other.     How  large  is  each  ? 


..     oi       f^-2/'  =  23-3a;?/.     ^    ^,       [23^-5^^  =  20- 


^1 


76.  A  farmer  has  two  small  fields,  each  an  exact  square.  It 
takes  200  rods  of  fence  to  inclose  both.  Together  they  contain 
8i  acres.     Find  the  dimensions  of  each. 

Simplify  and  express  with  positive  exponents : 

77.  ahhxah-h^.  gO     "^^x—- 

78.  (a'b'cf  X  (a^bciy.  3«  X  3"+^ 


^>s 


INDEX 

(Numbers  refer  to  Sections.) 


A-bsolute  value,  26. 

Addition,  14,  30,  39,  40,  103,  140. 

elimination  by,  126. 
Algebra,  1. 
Antecedent,  114. 
Arrangement  of  terms,  61. 
Axioms,  18. 

Base,  5. 
Binomial,  10. 
Braces,  7. 
Brackets,  7. 

Cancellation,  105. 
Checking  work,  41. 
Clearing  of  fractions,  111. 
Coefficient,  12. 

numerical,  12. 

literal,  12. 
Comparison,  elimination  by,  127. 
Complete  quadratic  equation,  144. 
Completing  the  square,  147. 
Complex  fractions,  109. 
Conditional  equations,  15. 
Consequent,  114. 
Constant,  120.    • 
Cubic  equation,  57. 

Degree  of  an  equation,  57. 

of  a  term,  57. 
Denominator,  97. 

lowest  common,  102. 
Distribution,  law  of,  50. 
Division,  35,  52,  53,  64,  56,  108,  142. 
Divisor,  trial,  135. 

true,  135. 

Elimination,  68. 
by  addition  or  subtraction,  126. 
by  comparison,  127. 
by  substitution,  128. 


Ellipse,  153. 
Equations,  15. 

complete  quadratic,  144. 

conditional,  15. 

cubic,  57. 

degree  of,  57. 

equivalent,  131. 

fractional,  110. 

graphs  of,  124,  153. 

identical,  15. 

inconsistent,  131. 

linear,  57. 

literal,  113. 

members  of,  15. 

pure  quadratic,  144. 

quadratic,  57. 

roots  of,  17. 

simultaneous,  66. 

systems  of,  67. 

to  solve,  17. 
Equivalent  equations,  131. 
Evaluation  of  polynomials,  11. 
Exponents,  5. 

fractional,  165. 

laws  of,  48,  52,  156,  157,  158,  159,  160 
162. 

negative,  163. 

positive  integral,  155. 

zero,  164. 
Expressions,  8. 

fractional,  100. 

integral,  100. 

literal,  8. 

mixed,  100. 

names  of,  10. 

number,  8. 
Extremes,  116. 

Factoring,  83. 

general  suggestions  on,  93. 
Factors.  ^ 
305 


B06 


INDEX 


Factors,  grouping  of,  47. 

highest  common,  94. 

order  of,  4li. 

prime,  83. 
Formula,  2, 

for  solving  quadratic  equations,  148. 
Fourth  proportional,  116. 
Fractional  equations,  110. 

exponents,  165. 

expressions,  100. 
Fractions,  i»7. 

clearing  equations  of,  111. 

complex,  109. 

denominator  of,  97. 

lowest  common  denominator  of,  102. 

numerator  of,  97. 

signs  of,  98. 

terms  of,  97. 

General  numbers,  3. 
Graphs,  29. 

of  linear  equations,  124. 

of  quadratic  equations,  153. 
Grouping,  laws  of,  30,  47. 

signs  of,  7. 

Highest  common  factor,  94. 

Identical  equations,  15. 
Identity,  15. 

Imaginary  numbers,  137. 
Inconsistent  equations,  131. 
Index  of  root,  72. 
Integral  expressions,  100. 

Law  of  distribution,  50. 

of  grouping  in  addition,  30. 

of  grouping  in  multiplication,  47. 

of  order  in  addition,  30. 

of  order  in  multiplication,  46. 
Laws  of  exponents,  48,  52,  156,  157,  158, 

159,  KJO,  162. 
Lever,  63. 
Like  terms,  13. 
Linear  equations,  57. 

graphs  of,  124. 
Literal  equations,  113. 

expressions,  8. 

numbers,  3. 
Lowest  common  denominator,  102. 
Lowest  common  multiple,  95. 


Mean  proportion,  116, 

proportional,  116. 
Means,  115. 

Members  of  an  equation,  15. 
Mixed  expressicms,  100. 
Monomial,  10. 
Multiple,  95. 

lowest  common,  95. 
Multiplication,  32,  49,  60,  51,  105,   106 
107,  141. 

signs  of,  4. 

Negative  exponents,  163. 

numbers,  23. 
Number  expression,  8. 
Numbers,  general,  3. 

imaginary,  137. 

literal,  3. 

negative,  23. 

opposite,  26. 

particular,  3. 

positive,  24. 

real,  137. 
Numerator,  97. 
Numerical  coefficient,  12. 

Opposite  numbers,  26. 
Order,  laws  of,  30, 46. 

Parabola,  153. 
Parentheses,  6. 
Particular  numbers,  3. 
Polynomials,  10. 

evaluation  of,  11. 
Positive  numbers,  24. 
Powers,  5. 

of  a  fraction,  160. 

of  a  monomial,  161. 

of  a  power,  158. 

of  a  product,  159. 

of   positive  and  negative    numbers^ 
34. 

roots  of,  162. 
Price  curves,  123. 
Prime  factors,  83. 
Proportion,  115, 

important  principles  in,  119. 

mean,  116. 

terras  of,  115. 
Proportional,  fourth,  116. 

mean,  116. 


INDEX 


307 


Proportional,  third,  116. 
Pure  quadratic  equation,  144. 

Quadratic  equation,  57. 

complete,  144. 

graph  of,  153. 

pure,  144. 
Quadratic  surd,  138. 
Quality,  signs  of,  25. 

Radical  sign,  72. 
Rate  of  motion,  61. 
Ratio,  114. 
Real  numbers,  137. 
Root,  index  of,  72. 

of  a  power,  162. 

of  an  equation,  17. 

square,  cube,  etc.,  72. 

Signs  of  a  fraction,  98. 

of  quality,  25. 
Signs  of  grouping,  7. 

insertion  of,  45. 

remo^  al  of,  44. 
Similar  terms,  13. 
Simple  machines,  118. 
Simultaneous  equations,  66. 
Solution  of  a  system,  67. 
Special  products  and  quotients,  70. 
Specific  gravity,  62. 
Speed,  61. 


Square  of  a  binomial,  78. 

of  a  polynomial,  80. 

of  arithmetical  numbers,  79. 

perfect,  86. 
Substitution,  20. 

elimination  by,  128, 152. 
Subtraction,  14,  31,  42,  43, 103, 140. 

elimination  by,  126, 
Surds,  138. 

quadratic,  138. 

simplest  form  of,  139. 
Systems  of  equations,  67. 

Terms,  9. 

arrangement  of,  51. 

degree  of,  57. 

of  a  fraction,  97. 

of  a  proportion,  115. 

similar,  13. 
Third  proportional,  116. 
Transposition,  36. 
Triangles,  similar,  117. 
Trinomial,  10. 
True  and  trial  divisors,  136L 

Variable,  120. 
Velocity,  61. 
Vinculum,  7. 

Zero  exponent,  164. 


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